Cobordism group with local coefficients and its application to 4-manifolds (Q5952193)

The oriented cobordism functor \(\{\Omega_{*} (X, A), \varphi_{*}, \partial\}\) satisfies the first six Eilenberg-Steenrod axioms for the category of pairs of topological spaces and maps [see \textit{P. E. Conner} and \textit{E. E. Floyd}, Differentiable periodic maps, Erg. Math. Gr. 33, Springer-Verlag, Berlin-Göttingen (1964)]. So for any CW complex \(X\) the Atiyah-Hirzebruch spectral sequence NEWLINE\[NEWLINE E^2_{p, q} = H_p(X; \Omega_q) \Rightarrow \Omega_{p + q}(X) NEWLINE\]NEWLINE is regular and hence convergent in the sense of \textit{H. Cartan} and \textit{S. Eilenberg} [Homological algebra, Princeton Math. Ser. 19, Princeton Univ. Press, Princeton (1956)]. Using this spectral sequence, closed oriented \(4\)-manifolds with finitely presentable fundamental group \(\pi\) can be classified stably (up to connected sums with simply connected closed \(4\)-manifolds) by the quotient \(H_4(B\pi; \mathbb Z)/(\operatorname {Aut} \pi)_{*}\) [see \textit{I. Hambleton} and \textit{M. Kreck}, in: Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lect. Note Ser. 197, 281-308 (1993; Zbl 0811.57007); \textit{I. Kurazono} and \textit{T. Matumoto}, Hiroshima Math. J. 28, No. 2, 207-212 (1998; Zbl 0911.57016)]. In the paper under review, the author extends the above result to the non-orientable case. For this, he introduces a cobordism group \(\Omega_n (X, A;\mathcal S_{w})\) with local coefficients for a pair \((X, A)\) of topological spaces and \(w \in H^1(X; \mathbb Z_2)\). This group reduces to \(\Omega_n(X, A)\) when \(w = 0\). If \(X\) is a CW complex and \(\mathcal S_{w}\) is a local system over \(X\) determined by \(w\), then there exists an Atiyah-Hirzebruch spectral sequence NEWLINE\[NEWLINE E^2_{p, q} = H_p(X; \Omega_q \otimes \mathcal S_{w}) \Rightarrow \Omega_{p + q}(X; \mathcal S_{w}) NEWLINE\]NEWLINE which is proved to be regular and hence convergent (see Theorem 1 of the paper). For a connected CW complex \(X\), there is a map NEWLINE\[NEWLINE \mu : \Omega_n(X; \mathcal S_{w}) \to H_n(X; \mathcal S_{w}) NEWLINE\]NEWLINE defined by \(\mu ([M, f, \varphi]) = f_{*}(\varphi_{*}(\sigma))\), where \(\sigma\) is a fundamental homology class in \(H_n(M; \mathcal S_{M})\) with respect to the orientation sheaf of a manifold \(M\) and \(\varphi\) is a local orientation (here \([M, f, \varphi]\) denotes a cobordism class in \(\Omega_n(X; \mathcal S_{w})\); for more details see the paper). The author proves that the map \(\mu : \Omega_4(X; \mathcal S_w) \to H_4(X; \mathcal S_w)\) is surjective and its kernel is \(\Omega_4\) if \(w = 0\), and \(\Omega_4 \otimes \mathbb Z_2\) if \(w \neq 0\) (see Corollary 2 of the paper). The main result of this fine paper is Theorem 3 where the closed \(4\)--manifolds with finitely generated abelian group \(\pi\) and first Stiefel--Whitney class induced from \(w\) (a non--trivial element of \(H^1(B\pi; \mathbb Z_2))\) are almost classified (up to connected sum with simply connected \(4\)--manifolds) by the quotient \(H_4(B\pi; \mathcal S_{w})/(\text{Aut} \pi)_{*}^{w}\) (via a one-to-one correspondence naturally induced by the above-mentioned map \(\mu\)).