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Cancellation for 4-manifolds with virtually abelian fundamental group. (English) Zbl 1369.57022

Two compact connected topological \(4\)-manifolds \(M_1\) and \(M_2\) are stably homeomorphic if there exists a homeomorphism between \(M_1\#k_1(S^2\times S^2)\) and \(M_2\#k_2(S^2\times S^2)\). The article proves the following cancellation result. Suppose \(\pi\) is a virtually abelian group and let \(n\) be such that \(\pi\) has a finite index, free abelian subgroup of rank \(n\). If \(M_1\) and \(M_2\) are stably homeomorphic and have the same Euler characteristic, then already \(M_1\#(n+2)(S^2\times S^2)\) and \(M_2\#(n+2)(S^2\times S^2)\) are homeomorphic.
A similar cancellation statement was proved by D. Crowley and J. Sixt [Forum Math. 23, No. 3, 483–538 (2011; Zbl 1243.57024)]. Here the bound for infinite groups is worse by one, but the result holds for all polycyclic-by-finite groups.
For finite groups, i.e.\(n=0\), it was proved by I. Hambleton and M. Kreck [J. Reine Angew. Math. 443, 21–47 (1993; Zbl 0779.57014)] that a stabilization with a single \(S^2\times S^2\) already suffices.
In the last section the homeomorphism classification of closed manifolds in the tangential homotopy type of \(X=X_-\#X_+\), where \(X_\pm\) are closed non-orientable topologicial 4-manifolds with order two fundamental group, is studied.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R65 Surgery and handlebodies
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