Cancellation for 4-manifolds with virtually abelian fundamental group

Abstract

Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $\pi$. For any $r \geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \# r(S^2 \times S^2)$ is homeomorphic to $X \# r(S^2\times S^2)$. How close is stable homeomorphism to homeomorphism? When the common fundamental group $\pi$ is virtually abelian, we show that large $r$ can be diminished to $n+2$, where $\pi$ has a finite-index subgroup that is free-abelian of rank $n$. In particular, if $\pi$ is finite then $n=0$, hence $X$ and $Y$ are $2$-stably homeomorphic, which is one $S^2 \times S^2$ summand in excess of the cancellation theorem of Hambleton--Kreck. The last section is a case-study investigation of the homeomorphism classification of closed manifolds in the tangential homotopy type of $X = X_- \# X_+$, where $X_\pm$ are closed nonorientable topological 4-manifolds whose fundamental groups have order two.Comment: 14 page

Embedded surfaces with infinite cyclic knot group

“…In general, λ NΣ 0 and λ NΣ 1 need not be isometric, even for N = D 4 , as shown by Oba [Oba17]. If as in (2) the Alexander polynomial is one and g + m is at least three, then we are able to leverage cancellation results for hyperbolic forms from [Bas73] (see also ([HT97,Kha17,MvdKV88,CS11]) to improve stable isometries to isometries.…”

“…The first step shows that the intersection forms are stably isometric after adding copies of H 2 . We translate this into a relative intersection form, and then apply a cancellation result of Bass [Bas73] (see also [MvdKV88], [HT97], [CS11, Corollary 7.7], and [Kha17]), given in Proposition 5.9. This allows us to deduce that λ NΣ i ∼ = Q X ⊕ H ⊕g 2 for i = 0, 1, and so λ NΣ 0 ∼ = λ NΣ 1 as desired.…”

Smooth structures on nonorientable four-manifolds and free involutions

“…Corollary 1 follows from [2], using the homeomorphism classification in [31,24]. Indeed, it is shown in [31,Theorem 2] that any closed smooth non-orientable 4manifold M with infinite cyclic fundamental group and ω 2 (M ) = 0 is stably homeomorphic to a connected sum of S 3 ×S 1 with copies of S 2 × S 2 .…”

Stably diffeomorphic manifolds and modified surgery obstructions

“…The literature on stably diffeomorphic manifolds contains work on the stable classification of 4-manifolds, that is enumerating and detecting the possible stable classes, including [CHR95, Spa03, KLPT17, KPT20]. There is also work showing that, under additional assumptions, a stable diffeomorphism type contains a unique diffeomorphism class [HKT09,CS11,Kha17]. In addition Davis gave conditions under which the homotopy type of a 4manifold M contains a unique stable class [Dav05].…”