Stable classification of 4-manifolds with 3-manifold fundamental groups

Abstract

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class.Comment: 55 pages, 2 figure

Homotopy classification of 4-manifolds whose fundamental group is dihedral

“…Recently, it was shown in [KPR20] that this is also true when the Sylow 2-subgroup of π 1 (X) is abelian with two generators, i.e. of the form Z/2 n × Z/2 m .…”

“…In this section, we will recall the basic theory of syzygies of finite groups. This offers an alternative perspective to some of the results which were discussed in [KPR20].…”

“…For a Zπ-module A, let Tors(A) denote the torsion subgroup of A as an abelian group. The following lemmas are elementary and we refer to [KPR20] for proofs.…”

The Kervaire-Milnor invariant in the stable classification of spin 4-manifolds

“…Relating the tertiary and Kervaire-Milnor invariants. We studied the primary invariant in [ KPT18 KPT18], and we studied the secondary and tertiary invariants in [KPT20 KPT20], building on [ KLPT17 KLPT17]. In [ KPT18 KPT18] and [KPT20 KPT20], we gave criteria which can decide whether (M, c) ∈ E 2,2 ⊆ Ω Spin 4 (Bπ), that is whether (M, c) is bordant to (M , c ) such that c : M → Bπ factors through the 2-skeleton Bπ (2) ⊆ Bπ.…”

Stable diffeomorphism classification of some unorientable 4-manifolds

“…This notion of equivalence has applications to quantum topology: for example, Reutter [Reu20, Theorem A] shows that the partition functions of 4d semisimple oriented TFTs are insensitive to stable diffeomorphism along the way to showing that such TFTs cannot distinguish homotopy-equivalent closed, oriented 4-manifolds. And stable diffeomorphism classes are computable: once the fundamental group G is fixed, Kreck [Kre99] shows how to reduce the classification of 4-manifolds up to stable diffeomorphism to a collection of bordism computations, and for many choices of G, the classification of closed, connected, oriented 4-manifolds with π 1 (M ) ∼ = G up to stable diffeomorphism has been completely worked out, thanks to work of Wall [Wal64], Teichner [Tei92], Spaggiari [Spa03], Crowley-Sixt [CS11], Politarczyk [Pol13], Kasprowski-Land-Powell-Teichner [KLPT17], Pedrotti [Ped17], Hambleton-Hildum [HH19], and Kasprowski-Powell-Teichner [KPT20].…”