Topological criteria for Floer $K_G$-split homology spheres and the K-theoretic route to the 11/8 conjecture

Manolescu (2014) associated to every integral homology 3-sphere $Y$ an invariant $\kappa(Y)$ defined from the Pin(2)-equivariant K-theory of the Seiberg-Witten Floer homotopy type of $Y$, and proved that $\kappa$ constrains the intersection forms of spin 4-manifolds bounded by $Y$. He introduced the class of Floer $K_G$-split homology spheres (those for which the relevant K-theoretic image ideal is of the form $(z^k)$) and proved that closed spin 4-manifolds admitting Bauer-type decompositions along $K_G$-split homology spheres cannot be counterexamples to Matsumoto's 11/8 conjecture, while explicitly asking for topological conditions guaranteeing $K_G$-splitness. Later work has shown that the source's conjecture $\kappa\ne\alpha$ has been resolved affirmatively (Dai-Sasahira-Stoffregen), that $\kappa$ has been computed for almost-rational plumbed, Seifert, and Dehn-surgery families, and that Hopkins-Lin-Shi-Xu proved a 10/8+4 theorem and showed that 10/8+4 is the limit of Furuta-Mahowald-class methods for the closed 11/8 conjecture. The surviving open core is the source question itself: find topological criteria for Floer $K_G$-splitness, the input that Manolescu's non-existence theorem for Bauer-type decompositions (published Theorem 4) requires; the Bauer-Manolescu decomposition strategy toward the 11/8 conjecture remains the motivation for the question.

Background

A smooth 4-manifold admits a spin structure when its second Stiefel-Whitney class vanishes; spin 4-manifolds have even intersection forms, i.e. the bilinear pairing on $H_2$ takes even values. By Donaldson's diagonalizability theorem a definite smooth intersection form must be diagonalizable, so an indefinite even unimodular intersection form is a sum $m(-E_8)\oplus nH$, where $E_8$ is the even unimodular positive-definite form of rank 8 and $H$ is the hyperbolic plane (the rank-2 even unimodular indefinite form). Matsumoto's 11/8 conjecture asserts that every closed, simply connected, smooth, spin 4-manifold $X$ satisfies $b_2(X)\ge\frac{11}{8}|\sigma(X)|$, where $b_2$ is the second Betti number and $\sigma$ the signature; in the above normal form this is $n\ge\frac{3m}{2}$. Furuta's 10/8 theorem, proved by finite-dimensional approximation of the Seiberg-Witten equations together with $\mathrm{Pin}(2)$-equivariant stable homotopy theory, gives the weaker bound $b_2\ge\frac{10}{8}|\sigma|+2$. The 11/8 conjecture remains open.

For manifolds with boundary an integral homology 3-sphere $Y$ (a closed 3-manifold with the homology of $S^3$; the Rokhlin invariant $\mu(Y)\in\mathbb{Z}/2$ is the signature mod 16 of any spin 4-manifold it bounds, divided by 8), the corresponding constraints involve the Seiberg-Witten Floer homotopy type $\mathrm{SWF}(Y)$. This is a pointed space with an action of the group $G=\mathrm{Pin}(2)$ (the double cover of $O(2)$, realized as $S^1\cup jS^1$ inside the unit quaternions), built as the homotopy type of Conley indices $I_\nu$ of finite-dimensional approximations of the Seiberg-Witten flow on $Y$; a Conley index is a pointed space encoding the isolated invariant set of a flow near a cutoff parameter $\nu$. Manolescu studied the $G$-equivariant K-theory of $I_\nu$: the inclusion of the $S^1$-fixed point set induces a map $\iota^*:\widetilde{K}_G(I_\nu)\to\widetilde{K}_G(I_\nu^{S^1})\cong\mathbb{Z}[w,z]/(w^2-2w,\ wz-2w)$, and $\kappa(Y)$ is defined from the smallest $k$ with $wx=2^k w$ for some $x$ in the image of $\iota^*$, after a metric-dependent normalization. He proved: $\kappa(Y)\bmod 2=\mu(Y)$; a definite-cobordism inequality; and for a spin cobordism $W$ from $Y_0$ to $Y_1$ with intersection form $p(-E_8)\oplus qH$, the bound $\kappa(Y_1)+q\ge\kappa(Y_0)+p-1$, strengthened to $\kappa(Y_1)+q\ge\kappa(Y_0)+p+1$ when $q>0$ and $Y_0$ is Floer $K_G$-split (arXiv Theorem 1.4; published as Theorem 3). For comparison, the invariant $\alpha(Y)$ is one of three integer invariants $\alpha,\beta,\gamma$ that Manolescu extracted from the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology, i.e. from the $\mathbb{F}_2$-Borel homology of $\mathrm{SWF}(Y)$; on all examples computed in the source, $\kappa=\alpha$, yet the paper conjectures $\kappa\ne\alpha$ in general, with the discrepancy attributed to possible 2-torsion at the end of the Borel homology tail.

Definition 1.3 of the source calls $Y$ Floer $K_G$-split if the image of $\iota^*$ is an ideal of $\mathbb{Z}[w,z]/(w^2-2w,\ wz-2w)$ of the form $(z^k)$, $k\ge0$; the three-sphere and the Brieskorn spheres $\pm\Sigma(2,3,12n+1)$ and $\pm\Sigma(2,3,12n+5)$ (Brieskorn spheres $\Sigma(a,b,c)$ are the links of the surface singularities $x^a+y^b+z^c=0$) are $K_G$-split, while $\pm\Sigma(2,3,12n-1)$ and $\pm\Sigma(2,3,12n-5)$ are not. The motivation is Bauer's proposed strategy for the 11/8 conjecture: a hypothetical counterexample with form $2r(-E_8)\oplus qH$, $q<3r$, stabilized to $q=3r-1$, can be cut (by a theorem of Freedman and Taylor on $\Lambda$-splitting, when $\pi_1=1$) into pieces of forms $2(-E_8)\oplus 3H$ and $2(-E_8)\oplus 2H$ glued along homology spheres $Y_i$. Manolescu's Theorem 4 (published numbering; arXiv Theorem 1.6) states that no closed four-manifold admits such a decomposition in which all the $Y_i$ are Floer $K_G$-split, and he writes that it would be worthwhile to find topological conditions guaranteeing $K_G$-splitness. Two later results frame what is now known: Hopkins-Lin-Shi-Xu completely determined the $\mathrm{Pin}(2)$-equivariant stable maps underlying Furuta's method, proving $b_2\ge\frac{10}{8}|\sigma|+4$ for all closed simply connected smooth spin 4-manifolds except $S^4$, $S^2\times S^2$, $K3$, and showing this 10/8+4 is the limit of such methods; Dai-Sasahira-Stoffregen computed $\mathrm{SWF}(Y)$ combinatorially via lattice homology for all almost-rational plumbed homology spheres (plumbed 3-manifolds whose plumbing graph becomes rational after reducing a single vertex weight, including all Seifert fibered spheres over $S^2$), which resolved $\kappa\ne\alpha$ - the two invariants diverge by arbitrarily much on connected sums $\#_n Y$ of suitable spheres - and determined $\kappa$ for that class, but did not produce topological criteria for $K_G$-splitness.

Problem Statement

Let $G=\mathrm{Pin}(2)$. For an integral homology 3-sphere $Y$, let $I_\nu$ be the Conley index of a finite-dimensional approximation of the Seiberg-Witten flow on $Y$ at a large cutoff $\nu$, and let $\iota^*:\widetilde{K}_G(I_\nu)\to\widetilde{K}_G(I_\nu^{S^1})\cong\mathbb{Z}[w,z]/(w^2-2w,\ wz-2w)$ be the restriction map in $G$-equivariant K-theory. Recall that $Y$ is Floer $K_G$-split if the image of $\iota^*$ is an ideal of the form $(z^k)$ for some $k\ge0$ (Manolescu, Definition 1.3). The problem is to find topological criteria for Floer $K_G$-splitness.

Find conditions on an integral homology 3-sphere $Y$, expressible and checkable in terms of classical or combinatorial data of $Y$ - for instance its plumbing graph, the Neumann-Siebenmann invariant $\bar\mu(Y)$ (the integer lift of the Rokhlin invariant for plumbed 3-manifolds), the monopole Frøyshov invariant $\delta(Y)$ (an integer from Seiberg-Witten Floer homology measuring the minimal grading shift admitted by any negative-definite spin filling of $Y$), Heegaard Floer correction terms (the rational-homology-cobordism-invariant rational numbers $d(Y,\mathfrak s)$ of Ozsvath-Szabo's Heegaard Floer homology), or the Pin(2)-equivariant local equivalence class (the equivalence class of $\mathrm{SWF}(Y)$ under maps inducing isomorphisms on Pin(2)-equivariant homology) - that imply, or ideally are equivalent to, $Y$ being Floer $K_G$-split, where the criterion does not presuppose the computation of the equivariant K-theory of $\mathrm{SWF}(Y)$. Currently only scattered examples are known: $S^3$, $\pm\Sigma(2,3,12n+1)$, and $\pm\Sigma(2,3,12n+5)$ are $K_G$-split, while $\pm\Sigma(2,3,12n-1)$ and $\pm\Sigma(2,3,12n-5)$ are not; for almost-rational plumbed spheres the value of $\kappa$ is now determined by the Dai-Sasahira-Stoffregen dichotomy: $\kappa=-\bar\mu$ when $-\bar\mu=\delta$ and $\kappa=-\bar\mu+2$ when $-\bar\mu<\delta$, but no general topological characterization of $K_G$-splitness is known.

As in the source, the motivation for this question is Manolescu's Theorem 4: any topological criterion for $K_G$-splitness is precisely the input that the Bauer-type decomposition strategy toward the 11/8 conjecture requires, since the obstruction applies exactly to decompositions along $K_G$-split boundaries. This motivation is background; the problem posed here is the source's criteria question itself.

References

1. Ciprian Manolescu, "On the intersection forms of spin four-manifolds with boundary", Mathematische Annalen 359 (2014), 695-728. DOI: 10.1007/s00208-014-1010-1; arXiv:1305.4667; https://arxiv.org/abs/1305.4667 (source paper)
2. Ciprian Manolescu, "Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture", Journal of the American Mathematical Society 29 (2016), no. 1, 147-176. DOI: 10.1090/jams829; arXiv:1303.2354; https://arxiv.org/abs/1303.2354
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