Counterexamples in 4-manifold topology

We illustrate the rich landscape of 4-manifold topology through the lens of counterexamples. We consider several of the most commonly studied equivalence relations on 4-manifolds and how they are related to one another. We explain implications e.g. that $h$-cobordant manifolds are stably homeomorphic, and we provide examples illustrating the failure of other potential implications. The information is conveniently organised in a flowchart and a table.


Introduction
The goal of this paper is to organise various equivalence relations in 4-manifold topology, and to understand the connections between them. We consider closed, connected 4-manifolds, unless otherwise specified, and we work in both the smooth and topological settings. Much work on 4-manifolds focusses on exotic behaviour, e.g. 4-manifolds that are homeomorphic but not diffeomorphic. We aim to illustrate, more broadly, the wealth of 4-manifold topology that has been discovered. The flowchart in Figure 1 shows the relationships between the equivalence relations we study. We will recall their definitions in Section 2, and prove the nontrivial implications in Section 3.  Figure 1. Equivalence relations on 4-manifolds. The implications shown in green are immediate. The black and blue implications are proven in Section 3. The blue implications hold when the corresponding condition is true, e.g. homotopy equivalent 4-manifolds are CP 2 -stably homeomorphic if their Kirby-Siebenmann invariants coincide. Where necessary for an implication to make sense, we assume that the manifolds are smooth. For example, the bottom arrow means that closed, smooth CP 2 -stably homeomorphic 4-manifolds are CP 2 -stably diffeomorphic, since the latter notion is only defined for smooth 4-manifolds.
It is straightforward to see that homeomorphism implies homotopy equivalence, for general spaces. We now explain a sequence of counterexamples to the converse for 4-manifolds, i.e. pairs of 4-manifolds that are homotopy equivalent but not homeomorphic. Along the way we illustrate our approach to investigating counterexamples. Namely, while investigating the failure of the converses of the implications in the flowchart, we will progressively impose restrictions on the counterexamples, e.g. that they be smooth, or orientable.
Conventions. We write Z/2 = {0, 1} for the integers modulo 2, a group under addition, and C 2 = {±1} for the cyclic group of order 2, with multiplication as the group operation. The symbol denotes homotopy equivalence, while s denotes simple homotopy equivalence. Depending on the context the symbol ∼ = denotes either homeomorphism or diffeomorphism.
Acknowledgements. The authors thank the Max Planck Institute for Mathematics in Bonn, where much of the research leading to this article occurred. We also thank Jim Davis, Kent Orr, Patrick Orson, and Peter Teichner for several helpful discussions. Finally we are very grateful to an anonymous referee for many helpful comments which helped us to improve the exposition.
DK was supported by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy -GZ 2047/1, Projekt-ID 390685813. MP was partially supported by EPSRC New Investigator grant EP/T028335/1 and EPSRC New Horizons grant EP/V04821X/1.

Equivalence relations on 4-manifolds
Recall that we implicitly assume throughout that 4-manifolds are closed and connected. We assume that the reader is familiar with homotopy equivalence, homeomorphism, and diffeomorphism of manifolds, and so we shall not define them. The classification of manifolds with respect to these three notions, and their comparison, is a central area of research. For example, the Poincaré conjecture, which has occupied topologists for over a century, asks for each n whether every homotopy equivalence from an n-manifold to the n-sphere is homotopic to a homeomorphism, or even to a diffeomorphism. 1 In dimensions at least five, surgery theory provides a concrete, effective framework within which one can try to improve a classification of manifolds up to homotopy equivalence to a classification up to homeomorphism or diffeomorphism. The programme can be applied to topological 4-manifolds under a restriction on the fundamental group; see Section 4 for an overview.
Next we discuss the various notions of stable equivalence.
Definition 2.1. The 4-manifolds M and N are said to be stably homeomorphic if there are integers s, t such that M # # s (S 2 × S 2 ) and N # # t (S 2 × S 2 ) are homeomorphic. They are said to be CP 2 -stably homeomorphic if there are integers s, t such that M # # s CP 2 and N # # t CP 2 are homeomorphic, for some choices of connected sum.
Definition 2.2. The smooth 4-manifolds M and N are said to be stably diffeomorphic if there are integers s, t such that M # # s (S 2 × S 2 ) and N # # t (S 2 × S 2 ) are diffeomorphic. They are said to be CP 2 -stably diffeomorphic if there are integers s, t such that M # # s CP 2 and N # # t CP 2 are diffeomorphic, for some choices of connected sum.
Note that S 2 × S 2 admits an orientation reversing self-diffeomorphism, so there is essentially only one choice of connected sum. On the other hand CP 2 does not admit any such diffeomorphism (nor homeomorphism), so for oriented manifolds there are two possible connected sums up to diffeomorphism/homeomorphism, usually denoted M #CP 2 and M #CP 2 . Therefore, the definitions above say that M and N are CP 2 -stably diffeomorphic (resp. homeomorphic) if there are integers s 1 , s 2 , t 1 , t 2 such that M # # s1 CP 2 # # s2 CP 2 and N # # t1 CP 2 # # t2 CP 2 are diffeomorphic (resp. homeomorphic). Note that for nonorientable manifolds, there is a unique connected sum N #CP 2 . We remark that some authors require s = t in the definition of stable homeomorphism and diffeomorphism. Manifolds M and N which are stably homeomorphic or diffeomorphic have s = t in our definition exactly when χ(M ) = χ(N ).
We emphasise that "stably" refers by default to connected sum with copies of S 2 × S 2 , and only "CP 2 -stably" refers to connected sum with copies of CP 2 .
To motivate Definitions 2.1 and 2.2, consider an alternative strategy to classify manifolds, based on Kreck's modified surgery [Kre99] and realised by Hambleton-Kreck for e.g. 4-manifolds with finite cyclic fundamental group in [HK88a,HK88b,HK93a,HK93b]: first classify manifolds up to stable homeomorphism, and then investigate the homeomorphism types within each stable class. In the latter step one attempts to prove that S 2 × S 2 summands can be cancelled, and this is what Hambleton and Kreck achieved for finite cyclic fundamental groups and also for more general finite groups under some additional hypotheses. This strategy in principle also applies to diffeomorphism classifications, but the cancellation step is much harder. Similarly, another approach to classification is to first classify manifolds up to CP 2 -stable equivalence, and then attempt to blow down extraneous CP 2 summands.
Definition 2.4. A homotopy equivalence f : X → Y between CW complexes X and Y is a simple homotopy equivalence if its Whitehead torsion τ (f ) vanishes. By Chapman's theorem [Cha74] the Whitehead torsion τ (f ) only depends on the homeomorphism type of X and Y . Hence Whitehead torsion is well-defined for homotopy equivalences between manifolds which are homeomorphic to CW complexes, e.g. smooth manifolds or closed manifolds of dimension = 4. It is an open question whether every topological 4-manifold is homeomorphic to a 4-dimensional CW complex. However, we can define the notion of simple homotopy equivalence of topological manifolds as follows. Embed M in high-dimensional Euclidean space. By [KS77, Essay III, Section 4], there is a normal disc bundle D(M ) admitting a triangulation. The inclusion map z M : M → D(M ) of the 0-section is a homotopy equivalence. Let z −1 M denote the homotopy inverse of z M .
Definition 2.5. We say that a homotopy equivalence f : M → N between topological manifolds (not necessarily of the same dimension) is a simple homotopy equivalence if the composition Definition 2.6. The 4-manifolds M and N are topologically s-cobordant if they cobound a topological h-cobordism W with trivial Whitehead torsion. The manifold W is called an s-cobordism. If in addition M , N , and W are smooth, then M and N are smoothly s-cobordant and W is called a smooth s-cobordism.
An h-or s-cobordism approximates a product, in the eyes of homotopy equivalence and simple homotopy equivalence respectively. One of the most spectacular results of the 20th century was Smale's h-cobordism theorem [Sma60,Sma61], which states that smooth, simply connected hcobordisms between n-manifolds with n ≥ 5 are indeed homeomorphic to products. This was later extended to other categories and to the case of s-cobordisms [Sma62, Bar63,Maz63,Sta67,KS77]. Consequences include the high-dimensional Poincaré conjecture in the piecewise-linear category in dimension at least five.
In dimension four, the celebrated work of Freedman and Quinn [Fre82,FQ90] includes an scobordism theorem, with a restriction on fundamental groups. This is the principal method for establishing the existence of a homeomorphism between 4-manifolds. We state the result in the next section as Theorem 3.5 and we outline the proof.

Justification of implications
The implications given in green in Figure 1 are immediate from the definitions. Now we justify the other implications.
Proof. Both statements follow from the diffeomorphism To establish the relationship between homotopy equivalence and CP 2 -stable homeomorphism, we will use the following theorem of Kreck. Recall that an identification of the fundamental group π 1 (M ) of a 4-manifold M with a group π determines a map c M : M → Bπ, up to homotopy, classifying the universal cover, where Bπ K(π, 1) is the classifying space.
Here Aut(π, w) denotes the set of automorphisms of π compatible with the map w. We have to factor out by the action of ± Aut(π, w) in order to account for the choice of identifications π 1 (M ) ∼ = π and π 1 (N ) ∼ = π, and for the choice of (twisted) fundamental classes in H 4 (−; Z w ).
The Kirby-Siebenmann invariant ks(M ) ∈ Z/2 of a 4-manifold M is by definition the unique obstruction for the stable tangent microbundle of M to admit a lift to a piecewise linear bundle. See [KS77,p. 318; FQ90, Section 10.2B; FNOP19, Section 8.2] for further details on the definition. In general it will suffice for us to know that the Kirby-Siebenmann invariant satisfies strong additivity properties, in particular under gluing and connected sum, and that it vanishes for a 4-manifold M if and only if M × R admits a smooth structure, if and only if M # # k S 2 × S 2 admits a smooth structure for some k.
Proof. For the first implication, let M and N be homotopy equivalent 4-manifolds with equal Kirby-Siebenmann invariants. Fix a map c N : N → Bπ as mentioned above the statement of Theorem 3.2 and let f : M → N be the claimed homotopy equivalence. Then define ∈ H 4 (π; Z w )/ ± Aut(π, w), and so by Theorem 3.2 (ii), we see that M and N are CP 2 -stably homeomorphic.
For the second implication, let M and N be CP 2 -stably homeomorphic and smooth. By Theorem 3.2 (ii) we see that c M * [M ] = c N * [N ] ∈ H 4 (π; Z w )/ ± Aut(π, w). Then apply Theorem 3.2 (i) to see that M and N are CP 2 -stably diffeomorphic.
Next we show the relationship between h-cobordism and stable diffeomorphism. The case of simply connected 4-manifolds was addressed by Wall in [Wal64b, Theorem 3]. A similar argument also applies in the general setting as explained by Lawson [Law78,Proposition]. We sketch the proof. Proof. The proof is the same in both cases, by using the fact that 5-dimensional topological cobordisms (W ; M, N ) admit handle decompositions, i.e. W can be built by attaching 5-dimensional handles to M × [0, 1] along topological embeddings of the attaching regions [Qui82, Theorem 2.3.1; FQ90, Theorem 9.1]. In the case of a smooth h-cobordism, we get a smooth handle decomposition relative to M by Morse theory.
In either case, we can perform handle trading to ensure that the handle decomposition has only 2-and 3-handles since the boundary inclusions M → W and N → W are 1-connected. Consider the middle level M 1/2 of the cobordism, obtained after attaching 2-handles to M . Since W is an h-cobordism, the 2-handles are attached along trivial circles, and so where t 1 + t 2 is the number of 2-handles. Here S 2 × S 2 is the twisted S 2 -bundle over S 2 .
If t 2 = 0 we are done. In case t 2 = 0, then there is an embedded 2-sphere in M 1/2 with odd framing of its normal bundle. Via the homotopy equivalence W → M , we see there is a map of a sphere to M with odd framing of its normal bundle, implying that the universal cover of M is nonspin. In this case we have that M #(S 2 × S 2 ) ∼ = M #(S 2 × S 2 ) (see e.g. [GS99, Exercise 5.2.6(b)]), and so we may assume t 2 = 0 again.
We have now argued that M 1/2 is a stabilisation of M . Applying the same argument to the upside down handlebody, we see further that M 1/2 is a stabilisation of N . Thus M and N are stably homeomorphic or diffeomorphic, depending on whether the handle decomposition was smooth or merely topological to begin with.
Next we discuss the s-cobordism theorem in dimension four. Below, we say a group is good if it satisfies the π 1 -null disc property [FT95] (see also [KOPR21a]). We do not repeat the definition here. In practice, it generally suffices to know that virtually solvable groups, and more generally groups of subexponential growth are good, and the class of good groups is closed under taking subgroups, quotients, extensions, and colimits [FT95,KQ00]. (2) For any ς ∈ Wh(π) there exists an h-cobordism (W ; M, M ) with τ (W ; M ) = ς.
(3) The function assigning to an h-cobordism (W ; M, M ) its Whitehead torsion τ (W ; M ) yields a bijection from the homeomorphism classes relative to M of h-cobordisms over M to the Whitehead group Wh(π).
Remark 3.6. It was asserted in [RS72,p. 90] that Theorem 3.5 (2) holds in dimension 4 in the piecewise-linear category, and without the assumption on π 1 (M ). But the proof there does not take into account the need for geometrically dual spheres to control the fundamental group of M .
Proof of Theorem 3.5.
admits a geometrically dual collection of (immersed) spheres {g i } k i=1 . Attach 3-handles to W along neighbourhoods of the spheres {f i } k i=1 to obtain a bordism W from M to M . Since we attached the handles of index 2 along trivial circles, π 1 (M ) ∼ = π 1 (W ). Since the {f i } have geometrically dual spheres, surgery along them does not change the fundamental group and so we have π 1 (M ) ∼ = π 1 (W ). The handle chain complex C * ( W , N ) is is an s-cobordism. Since π is good, W ∪ M W is homeomorphic to M × [0, 1] relative to M by (1). In particular N is homeomorphic to M . We can thus form (W ∪ M W ; M , M ), which again is an s-cobordism and is thus homeomorphic to M × [0, 1], relative to M by (1). We obtain a homeomorphism relative to M , as claimed.
Remark 3.7. Every 5-dimensional s-cobordism (W ; M, N ), with no restriction on fundamental groups, becomes homeomorphic to a product, relative to M , after sufficiently many connected sums with S 2 × S 2 × [0, 1] along arcs joining M and N [Qui83, Theorem 1.1]. This gives another proof of the special case of Theorem 3.4 that s-cobordant 4-manifolds are stably diffeomorphic/homeomorphic.
The last remaining implication in Figure 1 is given by the following result of Gompf.
In the same paper Gompf also showed that smoothings of a nonorientable 4-manifold become diffeomorphic after connected sum with sufficiently many copies of S 2 × S 2 .

Review of surgery exact sequences
In the next section we will appeal on several occasions to surgery exact sequences. We refer to [OPR21] for an account of the 4-dimensional case, and e.g. [Wal99] and [CLM19] for detailed treatments of general surgery theory in dimensions at least 5.
The surgery exact sequences are centred on the structure sets. For the remainder of this section, let M be a closed, connected, topological 4-manifold. Note that F is necessarily also a homotopy equivalence in the definition above. Note that F is necessarily a simple homotopy equivalence. Suppose that π 1 (M ) is a good group. Then every s-cobordism is a product, and we can alternatively describe the equivalence relation in the definition of the simple structure set without reference to s-cobordisms, by instead requiring a homeomorphism G : Continuing with the assumption that π 1 (M ) is good, it follows that one approach to the classification of manifolds simple homotopy equivalent to M , up to homeomorphism, is to first compute the simple structure set of M , and then to compute the set of orbits of the post- When π 1 (M ) is good, the surgery sequences are exact sequences of abelian groups, whose underlying sets are given as follows; the group structures arise via the theory of spectra and are hard to define geometrically [Nic82,Chapter 5]. The identity element of the structure sets is given by the identity map M → M . We will explain the terms other than the structure sets after stating the sequences. Let π := π 1 (M ) and let w : π → C 2 be the orientation character of M . Assume that π is a good group. For the homotopy structure set, we have an exact sequence: while for the simple structure set we have an exact sequence: The degree one normal maps, the terms involving N , are independent of the h and s decorations. For (X, ∂X) equal to either (M, ∅) or (M × [0, 1], M × {0, 1}), the set N (X, ∂X) consists of the set of manifolds (N, ∂N ) with a degree one map N → X that restricts to a homeomorphism on ∂N → ∂X, together with some normal bundle data that we will not define here, up to an analogous notion of degree one normal bordism. Details can be found in the references provided at the start of this section. It will suffice for us to know that (4.1) Similarly, for Y a closed 3-manifold, we have [Y, G/TOP] ∼ = H 2 (Y ; Z/2). As before the identity element in the set of normal maps is given by the identity map.
The L-groups L h 5 (Zπ, w), L s 5 (Zπ, w), L h 4 (Zπ, w), and L s 4 (Zπ, w) have purely algebraic definitions, in terms of π, w, and the decoration h or s. We will also not go into the details of the definitions here, but will give a brief overview.
Roughly speaking, L h 4 (Zπ, w) is defined in terms of nonsingular, sesquilinear, hermitian forms on free Zπ-modules. For L s 4 (Zπ, w) the Zπ-modules must be based, and certain isomorphisms between based modules are required to be simple, meaning that they represent the trivial element of the Whitehead group Wh(π). The identity element in the L 4 groups corresponds to the hyperbolic form, i.e. k ( 0 1 1 0 ) on ⊕ 2k Zπ for some k. There is also a version L p 4 (Zπ, w), where the underlying Zπ-modules are only required to be projective.
Elements of the L 5 groups consist of a hyperbolic form on a free Zπ-module equipped with a choice of a half rank summand of the base module, called a lagrangian, on which the form vanishes. For the s decorations, we need the module to be based and a certain short exact sequence related to the lagrangian to have trivial Whitehead torsion. The identity element of the L 5 -groups consists of a hyperbolic form where the lagrangian is standard. When the orientation character w is trivial, we often suppress w from the notation of L-groups.
The homomorphisms σ in the surgery sequences from degree one normal maps to the L-groups are called the surgery obstruction maps. Given a degree one normal map f : N → M , performing surgery on circles produces a map f , still with target M , that induces an isomorphism on fundamental groups. The element σ(f ) is given by the kernel of the map induced by f on second homotopy groups, called the surgery kernel. Exactness at N (M ) requires that π 1 (M ) is good and relies on [Fre82,FQ90].
Note that L 4 (Z) ∼ = Z, given by the signature of the form divided by 8. Hence σ : N (S 4 ) → L 4 (Z) sends [N, f ] to sign(N )/8. Using the naturality of the surgery exact sequence and that L 4 (Zπ) contains L 4 (Z) as a direct summand, we see that the summand H 4 (M ; )/8 and maps injectively to L 4 (Zπ). The preceding argument applies to L 4 (Zπ) with both h and s decorations.
The maps marked W in the surgery sequences are given by the Wall realisation actions of the L 5 groups on the structure sets, which we sketch next. Let h : N → M be a (simple) homotopy equivalence and let α be an element of the relevant L 5 group. Stabilising gives a map N # # k S 2 × S 2 → M , for some k, whose surgery kernel gives a hyperbolic form. We then represent the generators of the lagrangian in α by framed, disjointly embedded 2-spheres in N # # k S 2 × S 2 , on which we perform surgery. The resulting 4-manifold N comes equipped with a (simple) homotopy equivalence h : N → M and (N , h ) is by definition the element α · h of the structure set. By construction N and N are stably homeomorphic. It is highly nontrivial to represent the lagrangian by disjointly embedded spheres, and requires the work of [Fre82,FQ90] and the restriction to good fundamental groups. Figure 2 shows what we know about the converses of the implications in Figure 1. In this section we collect the counterexamples indicated in Figure 2, explaining their construction and properties. The properties are also collected in Table 1 at the end of the paper. 5.1. S 2 × S 2 and S 4 .

Counterexamples
• Both manifolds are smooth, orientable, and simply connected.
• As S 2 × S 2 ∼ = S 4 #(S 2 × S 2 ), they are stably diffeomorphic, and therefore stably homeomorphic and CP 2 -stably diffeomorphic. • χ(S 2 × S 2 ) = 4 = 2 = χ(S 4 ). Therefore, they are neither (simple) homotopy equivalent, s-nor h-cobordant (in either category), homeomorphic, nor diffeomorphic. This straightforward example shows that to meaningfully ask for homotopically inequivalent 4-manifolds that are stably diffeomorphic, one should also require that the Euler characteristics coincide. Equivalently, one should require that the number of copies of S 2 × S 2 added is the same for both manifolds, that is s = t in Definition 2.2. 5.2. S 2 × S 2 and S 2 × S 2 . As before S 2 × S 2 denotes the nontrivial S 2 -bundle over S 2 . It can be constructed by gluing two copies of D 2 × S 2 together using the Gluck twist on their common boundary S 1 × S 2 . Alternatively, recall that oriented 3-plane bundles over S 2 are classified up  , and there are two such homotopy classes. The nontrivial map gives a 3-plane bundle whose sphere bundle is S 2 ×S 2 .
• Both S 2 × S 2 and S 2 × S 2 are smooth, orientable, and simply connected.
This example shows that it is easy to find manifolds that are CP 2 -stably homeomorphic or CP 2 -stably diffeomorphic but do not satisfy any of the other equivalence relations. 5.3. CP 2 and Freedman's * CP 2 . Freedman [Fre82,p. 370] constructed the manifold * CP 2 , which he called the Chern manifold, as follows. Attach a 2-handle to D 4 along a +1-framed trefoil K, to obtain the 1-trace of the trefoil. The boundary is a homology sphere Σ, which bounds a compact, contractible manifold C [Fre82, Theorem 1.4'; FQ90, Corollary 9.3C]. Cap off the 1-trace with C. The resulting closed 4-manifold is * CP 2 . The same construction with any knot K with Arf(K) = 1 gives rise to a homeomorphic manifold.
• The manifolds CP 2 and * CP 2 are orientable and simply connected.
. Therefore, * CP 2 is not smoothable, even stably. • Since the manifolds have isometric intersection forms, they are homotopy equivalent [Whi49,Mil58] and have equal Euler characteristic χ(CP 2 ) = 3 = χ( * CP 2 ). Since the Whitehead group of the trivial group is trivial, the manifolds are also simple homotopy equivalent. • Since the Kirby-Siebenmann invariants are different, they are not CP 2 -stably homeomorphic and therefore not homeomorphic, not s-or h-cobordant, and not stably homeomorphic. • The smooth questions are not applicable to this pair. This example shows that one must restrict to manifolds with the same Kirby-Siebenmann invariant, and moreover ideally smooth manifolds, to find really interesting examples of homotopy equivalent but not homeomorphic manifolds. 5.4. RP 4 #CP 2 and R# * CP 2 . Here * CP 2 is the Chern manifold from Section 5.3. The manifold R was first constructed by Ruberman [Rub84], as follows. By [Fre82, Theorem 1.4'; FQ90, Corollary 9.3C], the Brieskorn sphere Σ(5, 9, 13) bounds a compact, contractible, topological 4manifold U. As a Seifert fibered manifold, Σ(5, 9, 13) admits an orientation preserving order two self-diffeomorphism t given by the antipodal map on the generic fibres. Since the parameters 5, 9, and 13 are all odd, t is also the antipodal map on the exceptional fibres and t has no fixed points. The manifold R is then defined as A similar construction was previously used by Fintushel-Stern [FS84] to construct a manifold homeomorphic, but not diffeomorphic to RP 4 , as we describe in Section 5.6. Ruberman's proof that R is not homeomorphic to RP 4 utilises Rochlin's theorem [Roh52] (see also [FK78; Kir89, Chapter XI]) and the fact that the Rochlin invariant µ(Σ(5, 9, 13)) = 1. That R is a homotopy RP 4 follows from the same principle as in [FS84]. Namely, write S 4 as U ∪ t U, and observe that R is the quotient under a free involution. The same construction can be applied to any integer homology sphere Σ admitting a free orientation preserving involution and with µ(Σ) = 1. While it is a priori not clear that the outcome is unique up to homeomorphism, this follows from the classification of closed, non-orientable 4-manifolds with order two fundamental group [HKT94, Theorem 3].
In the literature R is sometimes denoted * RP 4 . We prefer not to use this notation to avoid confusion with the star construction, defined in Section 5.11. The manifold R also arises via a surgery construction, which we outline after the following list.
• The manifolds RP 4 #CP 2 and R# * CP 2 are nonorientable with nontrivial fundamental group isomorphic to Z/2. • RP 4 #CP 2 is smooth by construction. Ruberman-Stern [RS97] showed that R# * CP 2 is also smoothable, as follows. First they showed that there exists a knot K with S 3 1 (K) = ∂X 1 (K) = Σ(5, 9, 13) a Brieskorn sphere, via an explicit Kirby calculus argument. Let t be a free orientation preserving involution of Σ(5, 9, 13) as described at the beginning of this section. Construct the smooth 4-manifold To see that X is homeomorphic to R# * CP 2 , let U K be the compact, contractible, topological 4-manifold with boundary Σ(5, 9, 13) provided by [Fre82, Theorem 1.4'; FQ90, Corollary 9.3C]. By Freedman [Fre82] and Boyer [Boy86], there is a homeomorphism * CP 2 #U K ∼ = (X 1 (K) ∪ Σ(5,9,13) U K )#U K ∼ = X 1 (K), since they are both compact, simply connected 4-manifolds with the same intersection form and the same integer homology sphere boundary. This homeomorphism descends to a homeomorphism * CP 2 #R ∼ = X when quotienting the boundaries by the involution. Alternatively, one can apply the classification theorem of [HKT94] to show that X is homeomorphic to R# * CP 2 . Different choices of K, and thereby the Brieskorn sphere, might give rise to different smooth structures on R# * CP 2 , but this is currently an open question.
• The manifolds RP 4 #CP 2 and R# * CP 2 are homotopy equivalent, and therefore have equal Euler characteristic χ(RP 4 #CP 2 ) = 2 = χ(R# * CP 2 ). Since the Whitehead group of Z/2 is trivial, they are also simple homotopy equivalent. • Since they are homotopy equivalent and smoothable, they are CP 2 -stably homeomorphic and CP 2 -stably diffeomorphic. • Hambleton-Kreck-Teichner [HKT94] showed that they are stably homeomorphic, but not homeomorphic. • The manifolds are stably diffeomorphic by [FNOP19, Theorem 12.3], which states that smooth, nonorientable, compact 4-manifolds with universal cover non-spin, that are stably homeomorphic, are stably diffeomorphic. • They are not s-cobordant, in either category, since if they were they would be homeomorphic by the topological s-cobordism theorem (Theorem 3.5). Since the Whitehead group of Z/2 is trivial, they are also not h-cobordant in either category.
We end this section by giving an alternative construction of the manifold R following Hambleton-Kreck-Teichner [HKT94, pp. 650-1]. There is a degree one normal map, namely the collapse map E 8 → S 4 , with domain the E 8 manifold constructed by Freedman [Fre82,Theorem 1.7]. Connect sum with RP 4 in both domain and codomain to obtain a degree one normal map The surgery kernel is the image of the E 8 form under the map L 4 (Z) → L 4 (Z[Z/2], w), where w : Z/2 → C 2 is the nontrivial character. But this is the zero map [Wal99, Chapter 13A, bottom of page 173]. By the definition of L 4 (Z[Z/2], w), this means that, perhaps after stabilising with copies of S 2 × S 2 , the surgery kernel is a hyperbolic form. Applying [PRT20, Corollary 1.4] to the surgery kernel, there is a homeomorphism for some integer k and for some 4-manifold X. By additivity of the Kirby-Siebenmann invariant (see e.g. [FNOP19, Theorem 8.2]) whereas RP 4 is smooth and so ks(RP 4 ) = 0. We can then define R := X.
By [HKT94, Theorem 3] the manifold X is determined up to homeomorphism by the existence of the homeomorphism (5.1), and in particular coincides with Ruberman's construction. Since we built R by killing the surgery kernel of a degree one normal map, we have constructed an element of the structure set, [R → RP 4 ] ∈ S(RP 4 ), and in particular R RP 4 . 5.5. Kreck's examples K3#RP 4 and # 11 (S 2 × S 2 )#RP 4 . These are a pair of relatively easy to understand exotic 4-manifolds discovered by Kreck [Kre84]. Indeed this was the first known exotic pair of closed 4-manifolds; the examples of Cappell-Shaneson discussed in Section 5.6 were constructed earlier, but they were not shown to be homeomorphic until much later [HKT94]. The K3 surface is a well-known smooth, simply connected 4-manifold. One way to construct it is to first consider E(1) := CP 2 # # 9 CP 2 , which comes with an elliptic fibration F : E(1) → S 2 (be warned that this is not a Serre or Hurewicz fibration). Generically the point inverse images are tori with trivial normal bundles T 2 × D 2 . The fibre sum of two copies of E(1) is the K3 surface: We can then construct the manifolds K3#RP 4 and # 11 (S 2 × S 2 )#RP 4 .
• Since they are homeomorphic, they are stably homeomorphic, CP 2 -stably homeomorphic, homotopy equivalent, simple homotopy equivalent, and topologically h-and s-cobordant. • Since they are not stably diffeomorphic, the two 4-manifolds are not diffeomorphic, and neither smoothly h-cobordant nor smoothly s-cobordant.
Remark 5.1. More generally, Kreck [Kre84, Theorem 1] showed that there is at least one such example of a pair of homeomorphic but not stably diffeomorphic smooth 4-manifolds for each 1-type (π, w : π → C 2 ) with π a finitely presented group and w nontrivial. By Gompf The last homeomorphism uses the classification of closed, simply connected 4-manifolds again.
Next we explain the obstruction to stable diffeomorphism. Heuristically, the construction of a homeomorphism above does not work smoothly, because one cannot split the K3 surface smoothly, because of Rochlin's theorem that smooth, spin, closed, 4-manifolds have signature divisible by 16 [Roh52] (see also [FK78; Kir89, Chapter XI]. When necessary we use RP ∞ BZ/2 = K(Z/2, 1) as a model for the Eilenberg-Maclane space. Define B := BZ/2 × BSpin and let ξ : B → BO be the composition where γ ⊥ is the orthogonal complement to the tautological line bundle γ over BZ/2 RP ∞ , p : BSpin → BO is the standard projection, and ⊕ is the map corresponding to the Whitney sum of stable bundles. Replace ξ by a fibration, and by an abuse of notation denote the resulting homotopy equivalent domain by the same letter B and the new map to BO again by ξ.
Let Ω 4 (B, ξ) denote the group of bordism classes of closed, smooth 4-manifolds, equipped with a lift to B of the classifying map ν M : M → BO of the stable normal bundle. In other words, Ω 4 (B, ξ) has elements represented by pairs (M, ν M ), where M is a closed, smooth 4-manifold and ν M classifies its stable normal bundle, such that the following diagram commutes.
Roughly speaking, this is because (B, ξ) was chosen to be compatible with the Stiefel-Whitney classes w 1 (ν M ) and w 2 (ν M ), and also with the fundamental group π 1 (M ). The following theorem is a special case of [Kre99, Theorem C].
Then M and N are stably diffeomorphic if and only if where hAut(B, ξ) denotes the group of fibre homotopy classes of fibre homotopy equivalences of the fibration ξ : B → BO.
Represent an element of Ω 4 (B, ξ) by a map f : M → RP 4 × BSpin, using the cellular approximation theorem and the fact that BZ/2 ∼ = RP ∞ . Let pr 1 : RP 4 × BSpin → RP 4 be the projection, and make pr 1 •f transverse to RP 3 ⊆ RP 4 . Let F ⊆ M be the inverse image f −1 (RP 3 × BSpin), which is a closed 3-manifold. The restriction of f to F determines, with a little work, an induced spin structure on F and a map F → RP 3 ⊆ RP ∞ ∼ = BZ/2. So F determines an element of Ω Spin 3 (BZ/2). Every element (F, f ) of Ω Spin 3 (BZ/2) bounds some spin 4-manifold W which admits a branched double covering W → W restricting to the double cover F → F corresponding to f with branching set a 2-dimensional submanifold Σ of W . We will show the existence of such a null-bordism in the next paragraph. Note that ∂(νF ) ∼ = F , where νF is the normal bundle of where Σ · Σ denotes the homological self-intersection number of Σ ⊆ W . Kreck showed that α is a well-defined invariant of Ω 4 (B, ξ), namely it is unaffected by cobordism over B and is independent of the choice of W and the branching set. This uses the Atiyah-Singer G-signature theorem [AS68, Section 6] (see also [Gor86]) and Rochlin's theorem [Roh52]  (BZ/2) ∼ = Z/8, generated by the pair consisting of (RP 3 , inc ×s : RP 3 → RP ∞ × BSpin), where inc is the inclusion and s is some choice of spin structure. The 3-manifold RP 3 bounds the 4-manifold V obtained by adding a 2-handle to D 4 along an unknot with framing coefficient 2. A generator of π 2 (V ) is represented by an embedded 2-sphere. Taking the 2-fold cover of V branched along such an embedding yields a branched covering V → V restricting to the standard nontrivial double cover S 3 ∼ = ∂ V → RP 3 ∼ = ∂V on the boundary. By taking boundary connected sums of V , we see that kRP 3 bounds a 4-manifold W k = k V with a branched double cover as claimed. For any element (F, f ) of Ω Spin 3 (BZ/2) we can now construct W := W k ∪ kRP 3 W , where W is a spin bordism from (F, f ) to kRP 3 over BZ/2. The null-bordism W admits a branched double cover as required in the definition of α. This completes the description of α.
Next we recall the construction of the Cappell-Shaneson R. Let The matrix A induces a diffeomorphism φ A : T 3 → T 3 . Consider the punctured 3-torus T 3 0 and the corresponding mapping torus for A, denoted M A,0 . Take RP 4 and remove a neighbourhood S 1 × D 3 of an embedded circle representing the generator of π 1 (RP 4 ) ∼ = Z/2. Then • The manifolds R and RP 4 are smooth and nonorientable. They have nontrivial fundamental group isomorphic to Z/2. • They have equal Euler characteristics χ(R) = 1 = χ(RP 4 ).
• Cappell-Shaneson [CS76b] showed that R and RP 4 are homotopy equivalent, and therefore simple homotopy equivalent. topologically h-and s-cobordant, CP 2 -stably homeomorphic and CP 2 -stably diffeomorphic. • Since they are not stably diffeomorphic they are not diffeomorphic. They are not smoothly h-or s-cobordant because they are not stably diffeomorphic. • As mentioned above, the Fintushel-Stern 4-manifold R F S has exactly the same properties. We now describe the diffeomorphism obstruction from [CS76b] used to show that R is not diffeomorphic to RP 4 . Consider the nonorientable linear S 3 -bundle over S 1 , which we denote by S 1 × S 3 . Cappell-Shaneson defined an invariant α : N (S 1 × S 3 ) → Z/32 from the set of smooth degree one normal maps with target S 1 × S 3 . Let f : R → S 1 × S 3 represent an element of N (S 1 × S 3 ). Define Σ R to be a framed 3-manifold f −1 (S 3 ) obtained by making f transverse to a fibre S 3 in S 1 × S 3 , taking the inverse image, and pulling back the framing. Let µ(Σ R ) ∈ Z/16 be the Rochlin invariant of Σ R , by definition the signature mod 16 of a framed 4manifold with boundary Σ R . In addition, write W := R \ νΣ R , and let σ(W ) ∈ Z be its signature. Then Cappell-Shaneson defined In [CS76b, Proposition 2.1] they showed that α is a well-defined map α : N (S 1 × S 3 ) → Z/32 as claimed. This invariant will be used again in Section 5.12; see also Kreck's invariant in Section 5.5, which was inspired by the Cappell-Shaneson invariant.
Let M A be the mapping torus for the diffeomorphism φ A : T 3 → T 3 used in the construction of R. In [CS76b, Propositions 2.1 and 2.2] it was shown that there is a degree one normal map f : M A → S 1 × S 3 and that α(M A , f ) is nonzero. As such, just as in Section 5.5, the exotic behaviour is due to Rochlin's theorem. In [CS76b, Theorem 3.1], Cappell and Shaneson use the nontriviality of α(M A , f ) to show that R is not diffeomorphic to RP 4 . This theorem shows that there is a homotopy equivalence h : R → RP 4 that is not homotopic to a diffeomorphism. Then since every homotopy equivalence of RP 4 is homotopic to the identity (cf. Proposition 5.3), it follows that there is no diffeomorphism between R and RP 4 . It was shown in [Gom91] that the double cover of R is diffeomorphic to S 4 . The Cappell-Shaneson construction can be varied by making different choices for the matrix A, giving rise to an exotic RP 4 denoted R A . The precise conditions are that A ∈ GL(3, Z) with det(A) = −1 and det(I − A 2 ) = ±1; call such a matrix a Cappell-Shaneson matrix. It can be seen from the construction that if A and A are similar Cappell-Shaneson matrices then R A and R A are diffeomorphic. The universal cover of R A is a smooth homotopy 4-sphere. Many of these have been shown to be diffeomorphic to S 4 -we refer to the introduction of [KY17] for a detailed survey. The most general result in this vein is that the universal cover of R A arising from a Cappell-Shaneson matrix A with trace n where −64 ≤ n ≤ 69 is diffeomorphic to S 4 [ibid.]. However, this corresponds to only finitely many similarity classes, and so there are infinitely many Cappell-Shaneson homotopy 4-spheres which remain as potential counter-examples to the smooth 4-dimensional Poincaré conjecture. 5.7. L × S 1 and L × S 1 . Let L and L be 3-dimensional lens spaces with the same fundamental group that are homotopy equivalent but not homeomorphic. These are of the form L p,q1 and L p,q2 with gcd(p, q i ) = 1, and 1 ≤ q i < p for i = 1, 2, such that for some n, q 1 q 2 ≡ ±n 2 mod p (for homotopy equivalent), and q i ≡ ±q ±1 j mod p (for non-homeomorphic) [Rei35]. See also [Bro60,DK01,Coh73].
• The manifolds L × S 1 and L × S 1 are smooth and orientable. They have fundamental group isomorphic to Z/p × Z, for some p. • They are homotopy equivalent because L and L are, and since they are smooth they are therefore CP 2 -stably homeomorphic and CP 2 -stably diffeomorphic. Then Of course this also reproves that they are not diffeomorphic. • The manifolds L × S 1 and L × S 1 are stably diffeomorphic, and therefore stably homeomorphic. We prove this in Proposition 5.3 below, by adapting a proof of Cappell-Shaneson [CS81]. That this is possible was stated in [Wei99]. • By the topological s-cobordism theorem (Theorem 3.5), L×S 1 and L ×S 1 are not topologically s-cobordant and are therefore not smoothly s-cobordant. Here we use that Z/p × Z is a good group. • They are not topologically h-cobordant, as we explain in Proposition 5.4 using [KS04]. It follows that they are not smoothly h-cobordant. • By choosing a large enough value of p, one may use the classification of lens spaces to find arbitrarily large, finite sets {L i × S 1 } i , such that the elements pairwise satisfy the above properties.
Proposition 5.3. Let L and L be 3-dimensional lens spaces that are homotopy equivalent but not homeomorphic. Then the 4-manifolds L × S 1 and L × S 1 are stably diffeomorphic.
Proof. The strategy is as follows. Let π 1 (L) ∼ = π 1 (L ) ∼ = Z/p and let h : L → L be a homotopy equivalence. Let S h (L ) be the homotopy structure set of L and consider the map in the surgery sequence η : S h (L ) → N (L ) ∼ = [L , G/TOP] with target the normal invariants of L . This map is defined for 3-manifolds, even though there is no analogue of the entire surgery sequence for 3-manifolds (but see [KT01, Theorem 4] for a version with a homology structure set). We will show that the homotopy equivalences h : L → L and Id : L → L determine equal elements η(h) = η(Id) ∈ N (L ) i.e. normally bordant degree one normal maps. Crossing with S 1 we see that η(h × Id) = η(Id × Id) ∈ N (L × S 1 ). Since Z/p × Z is a good group, the surgery sequence is exact. Therefore [h × Id : L × S 1 → L × S 1 ] and [Id × Id : L × S 1 → L × S 1 ] are in the same orbit of the action of L h 5 (Z[Z/p × Z]) on S h (L × S 1 ). Then, by the definition of the Wall realisation L h 5 action, it follows that L × S 1 and L × S 1 are stably homeomorphic. Then since these 4-manifolds are smooth and orientable, they are in fact stably diffeomorphic by Theorem 3.8. We therefore have to show that η(h) = η(Id) ∈ N (L ). For this we adapt the proof of [CS81, Proposition 2.1], where the corresponding fact for lens spaces with even order fundamental group was proven in all dimensions, under an additional hypothesis that the double covers are homeomorphic. We will show that in dimension 3 the extra hypotheses are not needed.
First recall that there is a 4-connected map k : G/TOP → K(Z/2, 2), corresponding to a universal cohomology class k ∈ H 2 (G/TOP; Z/2) [MM79,KT01]. This induces a homomorphism Consider η(h) and η(Id) as elements of [L , G/TOP]. Since η(Id) = 0, also k * (η(Id)) = 0 ∈ H 2 (L ; Z/2). So we have to show that k * (η(h)) = 0, and then we will have shown that both maps L → G/TOP are null-homotopic, and hence that the two lens spaces are normally bordant. If p is odd, then H 2 (L ; Z/2) ∼ = H 1 (L ; Z/2) = 0, so we are done. We therefore assume that p is even, in which case H 2 (L ; Z/2) = Z/2, and we have something to check. So let p = 2r, for some r ≥ 1. Now we diverge from the proof of [CS81]. Let f : L → L and f : L → L be the r-fold covers so that π 1 ( L) ∼ = Z/2 ∼ = π 1 ( L ). Note that L and L are again lens spaces, so L ∼ = L ∼ = RP 3 = L 2,1 . We claim that (f ) * : H 2 (L ; Z/2) → H 2 (RP 3 ; Z/2) is an isomorphism. Note that f * : H 1 (RP 3 ; Z) → H 1 (L ; Z) is given by multiplication with r. Hence on the cellular Z-chain complexes, f 1 is given by multiplication with r and thus f 2 is the identity as can be seen by considering the following commutative diagram of the cellular chain complexes over Z.
The claim that (f ) * : H 2 (L ; Z/2) → H 2 (RP 3 ; Z/2) is an isomorphism immediately follows from this. By the commutative square in order to prove k * (η(h)) = 0, it suffices to show that (f ) * (η(h)) = 0. Since (f ) * : N (L ) → N (RP 3 ) is given by pulling back along f , we have (f ) * (η(h)) = η( h), where h : RP 3 → RP 3 is obtained from lifting h • f : RP 3 → L to RP 3 along f as in the diagram We assert that every orientation-preserving homotopy self-equivalence of RP 3 , and so in particular h, is homotopic to the identity. It then follows that h is trivial in the structure set of RP 3 , and so η( h) = η(Id RP 3 ) = 0 as desired. It remains to prove the assertion that every orientation-preserving homotopy self-equivalence of RP 3 is homotopic to the identity. This can be proven via obstruction theory, by iteratively extending a map defined on RP 3 × {0, 1} to RP 3 × [0, 1]. Since the target RP 3 is path connnected, there is no obstruction to extending over the relative 1-cells of RP 3 × [0, 1], i.e. to defining the homotopy on the 0-cells of RP 3 . For k ≥ 2, the obstruction to extending over the relative k-cells of RP 3 × [0, 1] lies in H k (RP 3 × [0, 1], RP 3 × {0, 1}; π k−1 (RP 3 )).
For k = 2, the obstruction vanishes because both maps induce the identity on π 1 (RP 3 ) ∼ = Z/2. Since π 2 (RP 3 ) = π 2 (S 3 ) = 0, the remaining obstruction lies in The obstruction measures the difference in the degrees of the two maps. Since both are degree 1, the obstruction vanishes and the assertion is proved.
Proposition 5.4. Let L and L be homotopy equivalent lens spaces that are not homeomorphic. The manifolds L × S 1 and L × S 1 are not topologically h-cobordant.
Proof. Assume that there is a topological h-cobordism W from L × S 1 to L × S 1 . As in (5.3) above, taking a product with S 1 kills the Whitehead torsion, i.e. W × S 1 is an s-cobordism from L×S 1 ×S 1 to L ×S 1 ×S 1 . The high-dimensional s-cobordism theorem then implies that L×S 1 ×S 1 and L × S 1 × S 1 are homeomorphic. But this implies that L and L are homeomorphic by the toral stability property for lens spaces [KS04] which can be seen using higher ρ-invariants [Wei99]. This is a contradiction to our assumption on L and L . consisted of E(1) = CP 2 # # 9 CP 2 and the Dolgachev surface E(1) 2,3 , which is obtained from E(1) via two log transforms. Let us recall the construction. The 4-manifold E(1) admits the structure of an elliptic fibration f : E(1) → S 2 . Let T 2 ⊆ E(1) be a generic fibre. Its normal bundle is a copy of T 2 × D 2 embedded in E(1).
In general, a log transform is a surgery operation on a 4-manifold X with a smoothly embedded torus T 2 ⊆ X with trivial normal bundle which cuts out a neighbourhood T 2 × D 2 of T 2 , and glues it back via a diffeomorphism of ∂(T 2 × D 2 ) = T 3 . Let {α, β, [∂D 2 ]} be a basis for H 1 (T 3 ). Then by definition we reglue to form using a diffeomorphism ϕ p : T 3 → T 3 , for some p ∈ Z, corresponding to an element   1 0 0 0 0 1 0 −1 p   of GL(3, Z). Perform two of these log transform operations on E(1), on disjoint generic fibres of the elliptic fibration, one with p = 2 and one with p = 3. The resulting 4-manifold is the Dolgachev surface E(1) 2,3 . One can also construct E(1) 2,3 by a single knot surgery operation [FS98a; FS98b; GS99, Section 10.3; Akb16, Section 6.5] on a generic fibre of E(1), using a trefoil knot [Par04, p. 7; FS09, Lecture 6, Section 2]. Akbulut used this in [Akb12] to obtain a description of the Dolgachev surface without 1-or 3-handles.
• They have isometric intersection forms +1 ⊕ 9 −1 and so are homeomorphic [Fre82,FQ90], and therefore (simple) homotopy equivalent, stably homeomorphic, topologically h-and s-cobordant, and CP 2 -stably homeomorphic. • They are homeomorphic and orientable and therefore stably diffeomorphic and CP 2 -stably diffeomorphic (Theorem 3.8). • They are smoothly h-cobordant by [Wal64b, Theorem 2], and are therefore smoothly scobordant since the Whitehead group of the trivial group is trivial. • They are not diffeomorphic [Don87], via tools of Yang-Mills gauge theory. While Rochlin's theorem suffices to construct exotic pairs of nonorientable 4-manifolds (see Sections 5.5 and 5.6) it seems that one requires the full force of gauge theory to detect orientable exotic pairs. After Donaldson's work, Seiberg-Witten theory provided an easier, but nonetheless still highly nontrivial, way to distinguish manifolds such as E(1) and E(1) 2,3 which are related by log transforms or knot surgery. There is a large literature on generalisations of Donaldson's example, as described in [GS99,Akb16]. As stated before, since this is not our focus we restrict ourselves to recalling the first known example. 5.9. # 3 E 8 and the Leech manifold. Next we present 4-manifolds that are stably homeomorphic but not homotopy equivalent. Freedman [Fre82] showed that every nonsingular, symmetric, integral bilinear form can be realised as the intersection pairing of a closed, simply connected, topological 4-manifold. The forms ⊕ 3 E 8 and the Leech lattice are even, symmetric, positive definite bilinear forms of rank and signature 24, so they are realised by closed 4-manifolds that we denote by # 3 E 8 and Le respectively.
• The manifolds # 3 E 8 and Le are simply connected and orientable with χ( # 3 E 8 ) = 26 = χ(Le). • Since they are spin with signature 24, Rochlin's theorem implies that the manifolds both have nontrivial Kirby-Siebenmann invariant, and are therefore not smoothable. • The manifolds have inequivalent intersection pairings and are therefore not homotopy equivalent. As a result they are neither simple homotopy equivalent, homeomorphic, topologically h-cobordant, nor topologically s-cobordant.
• Since they are spin and the Euler characteristics and the signatures coincide, # 3 E 8 and Le are stably homeomorphic and therefore also CP 2 -stably homeomorphic, as follows. The stable classification of closed, simply connected, spin topological 4-manifolds is essentially due to Wall • The smooth questions are not applicable to this pair. The downside of this example is that the manifolds are not smoothable. It turns out that this is inevitable when considering simply connected 4-manifolds, as shown by the next proposition. In the definite case, the intersection forms must be diagonal by Donaldson's theorem [Don83], and so the intersection forms are isometric, and therefore the manifolds are homotopy equivalent [Whi49,Mil58]. For the indefinite case, note that since the hyperbolic form is even, and the intersection forms of M and N become isometric after stabilising, they are either both odd or both even. Indefinite forms are determined up to isometry by the rank, parity, and signature [MH73, Theorem 5.3] and so again M and N are homotopy equivalent.
Finally, we note that any pair of even, inequivalent, nonsingular, symmetric, integral, bilinear forms with equal rank and signature could have been used in this section to produce a pair of stably homeomorphic but not homotopy equivalent manifolds. 5.10. Kreck-Schafer manifolds. Kreck and Schafer [KS84] constructed smooth 4-manifolds that are stably diffeomorphic but not homotopy equivalent. As observed in Proposition 5.5, their examples are necessarily not simply connected.
They used the following general construction. Start with a finite presentation of a group. Form the corresponding presentation 2-complex X. Thicken it to a 5-dimensional manifold N (X), e.g. by embedding X in R 5 and letting N (X) denote a smooth regular neighbourhood [Wal66]. Then consider the 4-manifold ∂N (X). One can use this to find a 4-manifold with any given finitely presented fundamental group.
For any two finite 2-complexes X and X with the same fundamental group, there are integers m, n such that X ∨ m S 2 X ∨ n S 2 [HAM93,(40)]. It follows that the boundaries of the 5dimensional thickenings ∂N (X) and ∂N (X ) are stably diffeomorphic.
Kreck and Schafer used finite 2-complexes X and X with the same fundamental group, as above, that have the same Euler characteristic but are not homotopy equivalent. Finding examples of 2-complexes with this property is rather difficult, but examples are known [Met76,Sie77,Lus93]. Kreck and Schafer's obstruction applies for many nontrivial fundamental groups, the smallest of which is Z/5 × Z/5 × Z/5. Kreck and Schafer then showed that for their particular choices of X and X , the 4-manifolds ∂N (X) and ∂N (X ) are not homotopy equivalent. These manifolds have the additional interesting property that their intersection forms, and indeed their equivariant intersection forms, are hyperbolic.
• The manifolds ∂N (X) and ∂N (X ) are smooth, non-simply-connected, oriented manifolds with the same Euler characteristic that are stably diffeomorphic but not homotopy equivalent. • Since they are not homotopy equivalent, they are also not simple homotopy equivalent, nor homeomorphic, nor diffeomorphic, nor h-or s-cobordant in either category. • Since they are stably diffeomorphic they are stably homeomorphic, CP 2 -stably diffeomorphic, and CP 2 -stably homeomorphic. Kreck and Schafer found pairs of 4-manifolds with the properties listed. Are there stable diffeomorphism classes of smooth, oriented 4-manifolds containing infinitely many homotopy equivalence classes, all with the same Euler characteristic? Or even arbitrarily many? 5.11. Teichner's E#E and * E# * E. A star partner of a 4-manifold M is a manifold * M such that there exists a homeomorphism M # * CP 2 ∼ = * M #CP 2 preserving the decomposition on π 2 , where * CP 2 is the Chern manifold whose construction we recalled in Section 5.3. Let E denote the unique fibre bundle over RP 2 with fibre S 2 , that has orientable but not spin total space. We give a Kirby diagram in Figure 3. This is a smooth, closed, orientable 4-manifold with fundamental group Z/2. Teichner [Tei97] showed that E has a star partner * E which is simple homotopy equivalent to E but has opposite Kirby-Siebenmann invariant. This will also follow from the more general Lemma 5.7 and Proposition 5.11 below. By the surgery exact sequence, if π 1 (M ) ∼ = Z/2, then * M is unique up to homeomorphism if it exists (see also [Tei97,Theorem 1]). In particular, this means that * E is the unique star partner for E. 0 1 Figure 3. A Kirby diagram for the manifold E, the unique fibre bundle over RP 2 with fibre S 2 , that has orientable but not spin total space.
• The manifolds E#E and * E# * E are orientable, with nontrivial fundamental group isomorphic to Z/2 * Z/2. • The manifold E is smooth by construction, and therefore so is E#E. The manifold * E is not smoothable, but it is currently open whether * E# * E is smoothable. Indeed * E# * E has vanishing Kirby-Siebenmann invariant and so is stably smoothable, i.e. there exists a k such that * E# * E# # k (S 2 × S 2 ) is a smooth manifold.
• Teichner [Tei97, Proposition 3] showed that E#E and * E# * E are not stably homeomorphic, and therefore they are not homeomorphic and not s-or h-cobordant.
• They are simple homotopy equivalent, since E and * E are simple homotopy equivalent. This will also follow from Lemma 5.7 later, since we will show that they are both star partners of E# * E. Recall also that the Whitehead group of the infinite dihedral group is trivial [Sta65], and so it also suffices to know that they are homotopy equivalent. • Since they are homotopy equivalent and have trivial Kirby-Siebenmann invariant they are CP 2 -stably homeomorphic. The homeomorphism from (5.4) below gives an alternative argument, and shows that only one CP 2 -factor is required.
• For smooth examples with the same properties, consider the pair E#E# # k (S 2 × S 2 ) and * E# * E# # k (S 2 × S 2 ). As mentioned above, there exists k for which these are smooth since ks(E#E) = ks( * E# * E) = 0. These two manifolds are still simple homotopy equivalent, CP 2 -stably homeomorphic and therefore CP 2 -stably diffeomorphic, but not stably homeomorphic, and therefore not stably diffeomorphic, neither smoothly s-nor h-cobordant, and not diffeomorphic.
Taking connected sums of E#E and * E# * E, as well as sufficiently many copies of S 2 × S 2 , one can construct homotopy equivalence classes containing arbitrarily many stable homeomorphism classes of smooth, orientable 4-manifolds using the techniques of [Tei97]. We omit the details. In [Tei92b, Example 5.2.4], Teichner also constructed similar examples for finite fundamental groups with quaternionic 2-Sylow subgroup.
As a counterpoint to these examples we show that it is impossible to find infinite families of 4-manifolds that are all homotopy homotopy equivalent but pairwise not stably homeomorphic. For this proof, we will need the following terminology. The normal 1-type of a smooth 4-manifold M is a fibration ξ : B → BO, inducing an injection π 2 (B) → π 2 (BO) and an isomorphism on π i (B) → π i (BO) for i > 2, which further admits a lift ν M : M → B of the stable normal bundle ν M : M → BO, inducing an isomorphism π 1 (B) → π 1 (BO) and a surjection π 2 (B) → π 2 (BO). A choice of a lift ν M is called a normal 1-smoothing of M . For a normal 1-type (B, ξ), let Ω 4 (B, ξ) denote the group of bordism classes of normal 1-smoothings. For topological 4-manifolds, we have parallel notions of a topological normal 1-type B → BTOP and topological normal 1-smoothings lifting the stable topological normal bundle. For a topological normal 1-type (B, ξ), let Ω TOP 4 (B, ξ) denote the group of topological bordism classes of topological normal 1-smoothings.
Proposition 5.6. The set of stable homeomorphism types of closed 4-manifolds in a fixed homotopy type is finite. Moreover the set of stable diffeomorphism types of closed, smooth 4-manifolds in a fixed homotopy type is finite.
Proof. Let M be a closed 4-manifold with π := π 1 (M ) and orientation character w. We will use that the composition S h (M ) The first and last terms are finite sets, and therefore so is N PL (M ) ∼ = N Diff (M ), as desired.
Next we take the opportunity to prove some basic facts about the star construction, some of which were used in the discussion at the start of the section. We show that star partners are simple homotopy equivalent and that the star partnership relation is symmetric. Then we discuss uniqueness of star partners, and we give a criterion that guarantees star partners exist. Proof. We use an argument due to Stong [Sto94b, Section 2]. For a 4-manifold M and β ∈ π 2 (M ), let cap(M, β) be the result M ∪ β D 3 of adding a 3-cell to M along β. Let h : * M #CP 2 → M # * CP 2 be a homeomorphism preserving the decomposition of π 2 . Let α ∈ π 2 (CP 2 ) ∼ = Z be a generator. Then h * (α) generates π 2 ( * CP 2 ). Further cap(CP 2 , α) s S 4 and cap( * CP 2 , h * (α)) s S 4 . Therefore In the next proof we will need the following fact. If the universal cover M is non spin and the fundamental group of M is good, then * M is unique up to homeomorphism [Sto94b, Corollary 1.2].
Proposition 5.8. Let M be a 4-manifold with a star partner * M . Suppose that π 1 (M ) is good. Then the relation of being a star partner is symmetric, i.e. M is a star partner of * M .
Proof. We must show that there is a homeomorphism M #CP 2 ∼ = * M # * CP 2 , preserving the decomposition on π 2 .
Since * M is a star partner of M , we see that * M #CP 2 ∼ = M # * CP 2 , preserving the decomposition on π 2 . By taking a connected sum on both sides with * CP 2 , we see that * M # * CP 2 is a star partner of M # * CP 2 .
Next we give a general criterion for when star partners exist. For example, this can be used to establish the existence of * E. The proof will use the following notion.
Definition 5.10. Let M be a 4-manifold. An immersion of a 2-sphere α : S 2 M is said to be RP 2 -characteristic if for every immersion R : RP 2 M such that R * w 1 (M ) = 0, we have α · R ≡ R · R ∈ Z/2.
The following consequence of work of Stong [Sto94a] is probably well-known to the experts, but has not appeared in print before.
Proposition 5.11. Let M be a 4-manifold with good fundamental group and containing an immersion R : RP 2 M such that R · R ≡ 1 mod 2 and R * w 1 (M ) = 0. Then a star partner * M exists.
Proof. The manifold * M can be constructed as follows. Start with M # * CP 2 and let α be an immersed sphere in * CP 2 with trivial self-intersection number, µ(α) = 0, representing a generator of π 2 ( * CP 2 ). Note that α is self-dual. In the construction of * CP 2 from Section 5.3, one can find such an α by gluing together the track of a null-homotopy for the trefoil in D 4 with the core of the attached 2-handle, and then adjusting the self-intersection number by adding small cusps. Since the mod 2 intersection numbers are such that α·R = 0 = R·R ∈ Z/2, α is not RP 2 -characteristic. Stong [Sto94a, p. 1310] proved that in this setting, where π 1 (M ) is good, and α admits an algebraically dual immersed sphere but is not RP 2 -characteristic, then α is homotopic to an embedding α . Since λ(α , α ) = 1, it follows that α has a regular neighbourhood with boundary S 3 . Consequently, M # * CP 2 ∼ = N #CP 2 , where N is obtained from M # * CP 2 by replacing a regular neighbourhood of α by D 4 . By construction, N is a star partner for M , which we denote by * M . 5.12. Akbulut's exotic (S 1 × S 3 )#(S 2 × S 2 ). In [Akb85, §3; Akb88] (see also [Akb16, Section 9.5]), Akbulut constructed a smooth, closed 4-manifold P that is homotopy equivalent to Q := (S 1 × S 3 )#(S 2 × S 2 ), but not diffeomorphic to Q. We give the construction presently. A manifold with similar properties was first constructed by Akbulut in [Akb84]. We say more about that and other alternative constructions at the end of the section.
Let Σ denote the (2, 3, 7)-Brieskorn sphere. Recall that the Rochlin invariant of Σ, denoted by µ(Σ), is by definition the signature mod 16 of a smooth, spin 4-manifold with boundary Σ. In this case µ(Σ) ≡ 8 mod 16. Attach a nonorientable 1-handle to Σ × [0, 1], joining the two boundary components, followed by a pair of 2-handles, as shown in Figure 4. One then checks that the boundary of this new handlebody is S 3 , so it can be capped off with D 4 , yielding the desired manifold P .
• The manifolds P and Q are smooth, closed, and nonorientable. They have nontrivial fundamental group Z. • The Euler characteristics are equal: χ(P ) = 2 = χ(Q).
• The equivariant intersection form of P can be computed from its handle description.
Wang's classification [Wan95] of nonorientable 4-manifolds with fundamental group Z up to homeomorphism then shows that P and Q are homeomorphic, since they have isomorphic equivariant intersection forms and equal Kirby-Siebenmann invariant. Here P and Q both have trivial Kirby-Siebenmann invariant since they are smooth. • Since they are homeomorphic, P and Q are topologically h-cobordant, and consequently s-cobordant since the Whitehead group of Z is trivial. • They are not stably diffeomorphic, which implies that they are not smoothly h-cobordant; the latter fact was shown by Akbulut [Akb85,Theorem 3]. His proof can be adapted to show the stronger fact that P and Q are not stably diffeomorphic, as we explain next.
• Since P and Q are not smoothly h-cobordant, they not smoothly s-cobordant, nor diffeomorphic. • They are CP 2 -stably diffeomorphic and CP 2 -stably homeomorphic, because they are homotopy equivalent and both have vanishing Kirby-Siebenmann invariant. Moreover, via explicit handle manipulation [Akb85, Theorem 1; Akb88; Akb16, Exercise 9.3], one sees that P is the result of a Gluck twist on an embedded 2-sphere in Q. This shows that not only are P and Q CP 2 -stably diffeomorphic, but in fact P #CP 2 ∼ = Q#CP 2 [Akb85, Corollary 2].
Notably the pair P and Q comprise the first example where the Gluck twist operation on a 2-sphere changes the smooth structure of a 4-manifold. Whether this is possible in the orientable setting remains open. Other examples of the operation changing the smooth structure on nonorientable 4-manifolds are given in [Tor17; KPR22, Proposition 1.6]. See also [AY13] for a condition that implies the Gluck twist operation does not change the diffeomorphism type.
As mentioned above, another manifold with similar properties as P was constructed by Akbulut in [Akb84]. That manifold, which we call P , also has an explicit handle decomposition [Akb84, Figure 4.6] consisting of one 0-handle, one 1-handle, two 2-handles, one 3-handle, and one 4-handle. Since P is nonorientable, the 1-handle is necessarily nonorientable. In other words, P is obtained by attaching two 2-handles to S 1 ×D 3 and then capping off with another copy of S 1 ×D 3 . Akbulut showed using explicit moves on the handle decompositions that where P 0 := P \ Int S 1 ×D 3 and R is the Cappell-Shaneson exotic RP 4 from Section 5.6. The double cover of P is the standard (S 1 × S 3 )# # 2 S 2 × S 2 , as shown in [Tor17, Proposition 9], using the fact that the double cover of R is diffeomorphic to S 4 [Gom91]. Akbulut's construction from [Akb84] can be modified to use other Cappell-Shaneson RP 4 s, some of which are not known to have standard double covers. Akbulut showed that P is homotopy equivalent to Q. As with P , the classification result of Wang [Wan95] shows that P is homeomorphic to Q. Akbulut used explicit moves on the handle decompositions to show that P is not diffeomorphic to Q, reducing the problem to the fact that the Cappell-Shaneson exotic RP 4 from Section 5.6 is not diffeomorphic to RP 4 . In has been claimed [Akb85, Theorem 1; Akb88] that P and P are diffeomorphic, but a proof has so far not appeared.
Another construction of a manifold homeomorphic but not diffeomorphic to Q was given by Fintushel-Stern in [FS84], using the technology of [FS81]. By surgering an exceptional fibre of the (3, 5, 19)-Brieskorn sphere, they constructed K, a homology S 2 × S 1 . They then formed X, the mapping cylinder of the quotient map K → K/t where t is the free involution contained in the S 1 -action on K, as in Sections 5.4 and 5.6. To finish the construction Fintushel and Stern showed that K is the boundary of a homotopy (S 1 × D 3 )#(S 2 × S 2 ), whose union with X is the desired manifold M . Using the invariant defined in [FS81], they showed that M is not smoothly s-cobordant to Q. They also showed using the handle decomposition, and the fact that t is isotopic to the identity, that the double cover of M is diffeomorphic to the standard (S 1 ×S 3 )# # 2 S 2 ×S 2 .
It is not known whether Akbulut's P is diffeomorphic to M . 5.13. Kwasik-Schultz manifolds homotopy equivalent to L × S 1 . The existence of these manifolds is the content of Example 1.9 from the introduction, which was first proven in [KS04, Theorem 1.2]. We restate the theorem and give an original proof below.
Theorem 5.12. Let M := L × S 1 , where L is a lens space L p,q with p ≥ 2, 1 ≤ q < p, and (p, q) = 1. Then there is an infinite collection of closed, orientable, topological 4-manifolds {M i } ∞ i=1 , that are all simple homotopy equivalent to M but pairwise not homeomorphic.
Proof. The proof will use the simple surgery exact sequence. The simple L-group satisfies Since Z/p × Z is a good group, the simple surgery sequence is exact. The normal maps N (M ×  [BDK07], except that these manifolds are of course nonorientable.
We next prove the following lemma, which was used in the proof of Theorem 5.12.
Lemma 5.13. The group of homotopy self-equivalences hAut(M ) of M is a finite group. (B, M ) that we shall define below. Since π 2 (M ) = 0, the Postnikov 2-type of M is B := B(Z/p × Z). The homotopy classes of homotopy equivalences of B are therefore isomorphic to the automorphisms of the group Z/p × Z. This group of automorphisms is a finite group.
Let The four non-relative spin bordism groups, and the first and last maps in the sequence, can be computed using the (natural) Atiyah-Hirzebruch spectral sequence for the generalised homology theory of spin bordism. We omit the details, because very similar details will appear below in the proof of Lemma 5.14. There we will restrict to p a power of 2, but the computation that both the cokernel of the map Ω Spin (B) are finite groups is similar for all p ≥ 2. To avoid essentially repeating ourselves, we only give the details in the proof in Section 5.14 below, since in that case the result is new, and more precise upper bounds are required. It follows that Ω Spin 5 (B, M ) is finite. Therefore (5.5) shows that hAut(M ) is finite, as desired.
5.14. Simple homotopy equivalent, h-cobordant 4-manifolds that are not s-cobordant. We construct arbitrarily large collections of closed, orientable, topological 4-manifolds that are simple homotopy equivalent and h-cobordant but not topologically s-cobordant. This will prove Theorem 1.10, which we restate below. We will employ the same scheme as in the previous subsection. The manifolds in each collection will be simple homotopy equivalent and h-cobordant to a fixed 4-manifold L 2 r ,1 × S 1 , for some r. We will show that by making r large enough we can obtain a collection of 4-manifolds of any given size, with the following properties.
• They are orientable and have nontrivial fundamental group isomorphic to Z/2 r × Z for some r ≥ 8. • They are all simple homotopy equivalent and topologically h-cobordant to one another. As a result, they are homotopy equivalent, stably homeomorphic, and CP 2 -stably homeomorphic. They are are pairwise not topologically s-cobordant and therefore not homeomorphic. • Since they are homotopy equivalent they all have vanishing Euler characteristic, the same as L 2 r ,1 × S 1 . • They are all stably homeomorphic and CP 2 -stably homeomorphic.
• We do not know whether they are smoothable, and therefore the smooth questions are not applicable.
Theorem 1.10. For every n ≥ 1, there is a collection {N i } n i=1 of closed, orientable, topological 4-manifolds, that are all simple homotopy equivalent and h-cobordant to one another, but which are pairwise not s-cobordant.
Here we use that s = 0, in the notation of that theorem (this is a different s to the s-decoration of the L-groups L h 4 (ZG r ) = Z m(r) ⊕ (Z/2) n(r) where n(r) := 2(2 r−2 + 2)/3 − r/2 − 1. Putting this all together we have: L s 5 (Zπ r ) ∼ = Z m(r) ⊕ (Z/2) n(r) and L h 5 (Zπ r ) ∼ = Z m(r) . The kernel of the forgetful map L s 5 (Zπ r ) → L h 5 (Zπ r ) is the torsion summand K := ker(L s 5 (Zπ r ) → L h 5 (Zπ r )) ∼ = (Z/2) n(r) , since L h 5 (Zπ r ) is torsion free and the kernel of the map L s n (R) → L h n (R) is 2-torsion, for every ring with involution R and every n ∈ Z by the Rothenberg exact sequence [Sha69, Proposition 4.1]. The elements in K act on the simple structure set of M r , producing topological manifolds that are stably homeomorphic and simple homotopy equivalent to M r . We can compute the normal maps as We have a direct sum because there is a 5-connected map G/TOP → K(Z, 4) × K(Z/2, 2) [KT01]. The Z summand is detected by the ordinary signature, and in particular it maps to one of the multisignature summands in Z m(r) ⊆ L s 5 (Zπ r ), under the surgery obstruction map. The torsion summand Z/2 r ⊕ Z/2 could map to the torsion elements in L s 5 (Zπ r ). But at least a summand of L s 5 (Zπ r ), one isomorphic to (Z/2) n(r)−2 , acts nontrivially on the simple structure set. Note that |(Z/2) n(r)−2 | = 2 n(r)−2 .
Still, it might be the case that some of the elements of the simple structure set S s (M r ) obtained by this action of L s 5 are identified by the action of the simple homotopy self-equivalences hAut s (M r ). Note that | hAut s (M r )| ≤ | hAut(M r )|. We will show the following lemma.
Moreover, for any k there exists an r with 2 n(r)−2 /| hAut(M r )| > k.
Every element M r (κ) := κ · [Id : M r → M r ] ∈ S s (M r ) arising from the action of an element κ ∈ K maps trivially to the homotopy structure set S h (M r ), by the diagram The M r (κ) are all therefore h-cobordant to M r . Since they arise from the action of the simple Lgroup L s 5 (Zπ r ), they are all simple homotopy equivalent and stably homeomorphic to one another. By Lemma 5.14 there is more than one orbit of {M r (κ)} κ∈K / hAut s (M r ), and these are not scobordant and therefore not homeomorphic manifolds. Moreover, for a given k we can choose r so that there are at least k orbits, and therefore we find arbitrarily large collections. Now we prove Lemma 5.14.
Proof of Lemma 5.14. As in Section 5.13, we use the braid from [HK04]. There we claimed that hAut(L p,q × S 1 ) is finite for any lens space L p,q with p odd. Now we claim the same when p = 2 r , and moreover in this case we compute an explicit upper bound, in terms of r, for the order of hAut(L p,q × S 1 ). The braid includes the exact sequence [HK04, Corollary 2.13] Ω Spin 5 (B r , M r ) → hAut(M r ) → hAut(B r ), (5.6) so we need upper bounds for the cardinalities of Ω Spin 5 (B r , M r ) and hAut(B r ), where hAut(B r ) denotes the set of homotopy self-equivalences of the Postnikov 2-type B r up to homotopy. The spin bordism group Ω Spin 5 (B r , M r ) also appeared in the proof of Lemma 5.13, and we refer the reader there for the definition.
For an automorphism, we must have that z = ±1 and a must be a generator of Z/2 r . Hence there are 2 2r = 2 · 2 r−1 · 2 r allowed choices for z, a and b.
To find an upper bound for | Ω Spin We investigate the bordism groups using the Atiyah-Hirzebruch spectral sequence. The sequence we need, for X ∈ {M r , B r }, is E 2 p,q = H p (X; Ω Spin q ) ⇒ Ω Spin p+q (X). In the range of interest 0 ≤ q ≤ 5, we have: for q = 0, 4, Z/2, for q = 1, 2, 0, for q = 3, 5.
Additionally H 1 (M r ; Z) ∼ = Z/2 r ⊕ Z. Since B r = Bπ r can be constructed from M r by adding cells of dimension four and higher, for A ∈ {Z/2, Z} the induced map is an isomorphism for k = 0, 1, 2 and a surjection for k = 3. Finally we will need that H 5 (B r ; Z) ∼ = Z/2 r . Now we use this homology information together with the spectral sequences to obtain an upper bound for the cardinality of the cokernel of the map Ω Spin 5 (M r ) → Ω Spin 5 (B r ) from (5.7). The map M r → B r induces maps between each page of the spectral sequences computing Ω Spin 5 (M r ) and Ω Spin 5 (B r ). The nonzero terms E 2 p,q on the E 2 page with p + q = 5 are as follows: H 1 (−; Z), H 3 (−; Z/2), H 4 (−; Z/2), and H 5 (−; Z).
The maps H 1 (M r ; Z) → H 1 (B r ; Z) and H 3 (M r ; Z/2) → H 3 (M r ; Z/2) are onto as explained above, so by naturality of the spectral sequence these terms do not contribute to the cokernel. The mod 2 fundamental class in H 4 (M r ; Z/2) ∼ = Z/2 maps nontrivially to H 4 (B r ; Z/2) ∼ = (Z/2) 2 , so possibly one Z/2 could contribute to the cokernel (whether or not it does so depends on differentials which we shall not take into account). The only other contribution to the cokernel comes from the term H 5 (B r ; Z) ∼ = Z/2 r . As a result the cokernel of Ω Spin It follows that the kernel of Ω Spin 4 (M r ) → Ω Spin 4 (B r ) is generated by elements coming from the terms H 2 (M r ; Z/2) ∼ = (Z/2) 2 and H 3 (M r ; Z/2) ∼ = (Z/2) 2 , and so the kernel has at most 2 4 elements. Thus by (5.7), we see that | Ω Spin 5 (B r , M r )| ≤ 2 r+1 · 2 4 = 2 r+5 .