In this article we study the problem of minimizing aχ + bσ on the class of all symplectic 4-manifolds with prescribed fundamental group G (χ is the Euler characteristic, σ is the signature, and a, b ∈ ℝ), focusing on the important cases χ, χ + σ and 2χ + 3σ. In certain situations we can derive lower bounds for these functions and describe symplectic 4-manifolds which are minimizers. We derive an upper bound for the minimum of χ and χ + σ in terms of the presentation of G.
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Scott Baldridge, Paul Kirk, On symplectic 4-manifolds with prescribed fundamental group. Comment. Math. Helv. 82 (2007), no. 4, pp. 845–875DOI 10.4171/CMH/112