Surgery on closed 4-manifolds with free fundamental group

Abstract

The 4-dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class of canonical problems with free fundamental group [5, chapter 12]. The proof of the conjecture for ‘good’ groups relies on the disk embedding theorem (see [5]), which is not presently known to hold for arbitrary groups. However, in certain cases it may be shown that surgery works even when the disk embedding theorem is not available for a given fundamental group (such results still use the disk-embedding theorem in the simply-connected setting, proved in [3]). For example, this is possible when the surgery kernel is represented by π1-null spheres [4], or more generally by a π1-null submanifold satisfying a certain condition on Dwyer's filtration on second homology [7]. Here we state another instance when the surgery conjecture holds for free groups. The following results are stated in the topological category.

On smoothable surgery for 4–manifolds

“…Indeed, taking all ‫ޅ‬ m i D ‫,ޒ‬ we obtain the finite-rank free groups D F n . Thus we partially strengthen a theorem of Krushkal and Lee [17] if X is a compact, connected, oriented TOP 4-manifold with fundamental group F n . They only required X to be a finite Poincaré complex of dimension 4 (@X D ¿) but insisted that the intersection form over ‫ޚ‬OE of their degree one, TOP normal maps f W M !…”

Families of 4-Manifolds with Nontrivial Stable Cohomotopy Seiberg–Witten Invariants, and Normalized Ricci Flow

“…Theorem 12 (Hambleton-Teichner [25], see also [39].) Let X be a smooth closed oriented 4-manifold with infinite cyclic fundamental group.…”

Surgery and involutions on 4–manifolds

“…Definition 3.1 In the general context (not assuming the existence of an involution on N ) we say that a homotopy equivalence f : N −→ Γ is weakly equivariant if the restriction of f to ∂N is equivariant with respect to the obvious Z/2 action, and there is a diffeomorphism from Note that in the analogous context: given a homotopy equivalence f from a closed 4-manifold N to a graph, the point inverses may be arranged to be 3-spheres [9], up to an s-cobordism of N . (Changing N by an s-cobordism is fine for the applications to the surgery conjecture.)…”

“…Note that in the analogous context: given a homotopy equivalence f from a closed 4-manifold N to a graph, the point inverses may be arranged to be 3spheres [9], up to an s-cobordism of N . (Changing N by an s-cobordism is fine for the applications to the surgery conjecture.)…”