In the 2-jet space J_2(n, p) of smooth map germs (ℝ_n, 0) → (ℝ_p_, 0) with n ≥ p ≥ 2, we consider the subspace Ω_n−p_+1,0(n, p) consisting of all 2-jets of regular germs and map germs with fold singularities. In this paper we determine the homotopy type of the space Ω_n−p_+1,0(n, p). Let N and P be smooth (C_∞) manifolds of dimensions n and p. A smooth map f : N → P is called a fold-map if f has only fold singularities. We will prove that this homotopy type is very useful in ﬁnding invariants of fold-maps. For instance, by applying the homotopy principle for fold-maps in [An3] and [An4] we prove that if n − p + 1 is odd and P is connected, then there exists a surjection of the set of cobordism classes of fold-maps into P to the stable homotopy group lim_k,l →∞ π__n+k+l(T(νPk) ∧ T(γ^l__G__n−p+1,l)). Here, νPk is the normal bundle of P in ℝ_p_+k and γ^l__G__n−p+1,l) denote the canonical vector bundles of dimension l over the grassman manifold G__n−p+1,l. We also prove the oriented version.