Definite spin and $E_8$-boundings of homology 3-spheres: Tange's invariants $\mathfrak{ds}$, $g_8$, and the (11/8)-linked questions

An integral homology 3-sphere $Y$ may or may not bound a smooth, homologically 1-connected, definite (positive- or negative-definite), spin 4-manifold; when it does, the intersection form is an even unimodular definite lattice — a multiple of the $E_8$ lattice in rank 8, but in higher rank also lattices such as $D_{16}^+$ or the Niemeier lattices. Tange (Topology and its Applications 202 (2016) 160–182) introduced the invariants $\mathfrak{ds}(Y)\ge\overline{\mathfrak{ds}}(Y)$, the maximum and minimum of $b_2(X)/8$ over such boundings, and the $E_8$-genera $g_8(Y)\ge\overline{g_8}(Y)$, the maximum and minimum multiplicity of the $E_8$ form; he computed them for infinite Brieskorn families and posed a consolidated list of open questions: which homology spheres admit definite spin boundings; whether finiteness of $\mathfrak{ds}$ forces an $E_8$-bounding; when the sharp comparisons with the Ozsváth–Szabó correction term $d$, the Neumann–Siebenmann invariant $\bar\mu$, and the Bohr–Lee invariants $m,\overline m$ hold; bounding-genus inequalities for explicit Brieskorn families equivalent to instances of Matsumoto's (11/8)-conjecture; the constructive problem of producing $E_8$-boundings; additivity under connected sum; and an a priori bound on the fiber-minus-section classes $k[f]-[s]$ in elliptic surfaces $E(n)$ representable by embedded 2-spheres. One strict-inequality facet was resolved affirmatively by Aceto–Golla–Larson (2017); the characterization, (11/8)-linked, constructive, and additivity questions remain open.

Background

Let $Y$ be an integral homology 3-sphere, i.e. a closed oriented 3-manifold with the same integer homology as $S^3$. Every 3-manifold bounds smooth spin 4-manifolds, but whether $Y$ bounds one whose intersection form $Q_X$ (the unimodular symmetric bilinear form on $H_2(X;\mathbb{Z})$) is definite, meaning $b_2(X)=|\sigma(X)|$, is a subtle question controlled by gauge theory. A spin structure ($w_2(X)=0$) forces $Q_X$ to be even (all self-intersections even), so the intersection form of a homologically 1-connected (connected with $H_1(X)=0$) definite spin bounding of a homology sphere is an even definite unimodular lattice, whose rank is divisible by 8. Such lattices are not unique beyond rank 8: besides the $E_8$ lattice (unique in rank 8), rank 16 also admits $D_{16}^+$, rank 24 admits the 24 Niemeier lattices, and the number grows rapidly with rank. Following Tange, a definite spin bounding whose form is a multiple $\pm nE_8$ is called an $E_8$-bounding.

The classical constraints are: Donaldson's diagonalization theorem (a closed smooth definite 4-manifold has diagonal form, so no closed definite form can be $E_8$; it also implies a homology sphere cannot admit both a positive- and a negative-definite spin bounding, which makes Tange's sign invariant $\epsilon(Y)\in\{1,-1,0,\infty\}$ well defined); the Ozsváth–Szabó correction term $d(Y)$, a $\mathbb{Z}$-valued homology-cobordism invariant from Heegaard Floer homology, which for a spin negative-definite bounding $X$ gives $b_2(X)\le 4d(Y)$; the Neumann–Siebenmann invariant $\bar\mu$ of Seifert homology spheres, for which Ue proved $b_2(X)\equiv -8\bar\mu(Y)\pmod{16}$ together with explicit rank bounds; and Matsumoto's (11/8)-conjecture — still open — which states that a closed spin 4-manifold with intersection form $2k(-E_8)\oplus lH$ (with $H$ the hyperbolic plane) must satisfy $l\ge 3|k|$; Furuta's 10/8 theorem, which gives only $b_2(X)\ge\frac{10}{8}|\sigma(X)|+2$ (i.e. $l\ge 2|k|+1$ in this normalization), remains the best general result toward it.

Tange (2016) introduced, for a homology 3-sphere $Y$, the invariants $\mathfrak{ds}(Y)$ and $\overline{\mathfrak{ds}}(Y)$ — the maximum and minimum of $b_2(X)/8$ over homologically 1-connected definite spin boundings — and the $E_8$-genera $g_8(Y)$ and $\overline{g_8}(Y)$ — the maximum and minimum $|n|$ over $E_8$-boundings with $Q_X=nE_8$ (each set to $+\infty$ when no such bounding exists; note the unbarred variant is the maximum). Furuta's inequality gives $\mathfrak{ds}(Y)\le 9\overline{\mathfrak{ds}}(Y)+8$ whenever $\overline{\mathfrak{ds}}(Y)<\mathfrak{ds}(Y)$, and $|Y|\ge\mathfrak{ds}(Y)+1$ whenever $\mathfrak{ds}(Y)$ is even, where $|Y|=\min\{n\mid \partial X=Y,\ Q_X=nH\}$ is the bounding genus, defined when the Rokhlin invariant $\mu(Y)=\sigma(W)/8\bmod 2$ (for any spin bounding $W$) vanishes; the conjectured inequality $|Y|\ge\frac{3}{2}\mathfrak{ds}(Y)$ is equivalent to the (11/8)-conjecture. The paper also relates $\mathfrak{ds}$ to the Bohr–Lee invariants, $m(Y)$ and $\overline m(Y)$, defined as the maximum and minimum of $\frac54\sigma(X)-b_2(X)$ over spin boundings $X$ of $Y$ (with $\sigma$ the signature). Among its computations: $\mathfrak{ds}=n$ for the four Brieskorn families $\Sigma(4n-2,4n-1,8n-3)$, $\Sigma(4n-1,4n,8n-1)$, $\Sigma(4n-2,4n-1,8n^2-4n+1)$, and $\Sigma(4n-1,4n,8n^2-1)$, where $\Sigma(a,b,c)$ is the Brieskorn homology sphere given as the link of the complex surface singularity $x^a+y^b+z^c=0$; $g_8=1$ for $M_n=\Sigma(2,3,6n-1)\#(-\Sigma(2,3,6n-5))$; and $g_8(Y^-_{2k+1})=1$ for $0\le k\le 12$ and $k=14$, for the family of Brieskorn spheres $Y^-_n:=\Sigma(2,3,6n-1)$ (so $Y^-_1=\Sigma(2,3,5)$, the Poincaré sphere). These last $E_8$-boundings come from explicit handle decompositions embedded in the elliptic surface $E(1)$ (the complex surface fibered over $S^2$ with torus fibers and 12 singular fibers, diffeomorphic to $\mathbb{C}P^2\#9\overline{\mathbb{C}P}^2$), with complements the standard simply connected pieces known as Gompf nuclei; as a by-product, the class $a_k=k[f]-[s]$ (fiber class minus section class, $a_k^2=-2k-1$) is represented by a smoothly embedded 2-sphere in $E(1)$ for those same $k$. The paper compares this with Finashin–Mikhalkin's theorem that $E(2)$, the K3 surface, contains an embedded sphere of square $-86$, which implies $k[f]-[s]\in H_2(E(2))$ is spherical for $k\le 42$, and with Li–Li's computation of smooth minimal genera for all classes of square $>-(n+7)$ in $\mathbb{C}P^2\#n\overline{\mathbb{C}P}^2$ for $n\le 9$. Section 5 of the paper (together with Question 3.1) then poses the consolidated question list stated below.

Problem Statement

The problem is the consolidated research program posed in Section 5 (with its auxiliary Question 3.1) of Tange's paper. The common objective is to characterize the definite spin and $E_8$-boundings of integral homology 3-spheres — existence, extremal sizes, comparison with the classical gauge-theoretic invariants, behavior under connected sum — together with the constructive question of producing such boundings and the embedded-sphere question extracted from the paper's constructions. The numbered questions below are independent. Numbering follows the published version (arXiv numbers in parentheses where they differ):

(Q5.1) For an integral homology 3-sphere $Y$: (1) determine when $Y$ admits a homologically 1-connected definite spin bounding, i.e. when $\mathfrak{ds}(Y)<\infty$; (2) if $\mathfrak{ds}(Y)<\infty$, must $Y$ admit an $E_8$-bounding, i.e. can the bounding always be chosen with intersection form a multiple of $\pm E_8$ rather than another even definite unimodular lattice such as $D_{16}^+$ or a Niemeier lattice? (3) determine when the equalities $m(-Y)/2=\mathfrak{ds}(Y)$ and $\overline{m}(-Y)/2=\overline{\mathfrak{ds}}(Y)$ hold for the Bohr–Lee invariants $m,\overline m$.

(Q5.2, remaining part) Does there exist a homology 3-sphere $Y$ with $\mathfrak{ds}(Y)\ne g_8(Y)$ or $\overline{\mathfrak{ds}}(Y)\ne\overline{g_8}(Y)$ — that is, a definite spin bounding whose lattice is not an $E_8$-multiple, with no $E_8$-bounding of matching size? (The companion strict-inequality question — whether $\overline{g_8}(Y)<g_8(Y)$ can occur — was answered affirmatively by Aceto–Golla–Larson and is no longer part of what remains.)

(Q5.3) For a Brieskorn homology 3-sphere $Y$ with $4d(Y)=-8\bar\mu(Y)>0$: does the maximal even definite spin bounding attain the Ozsváth–Szabó bound, i.e. is $8\,\mathfrak{ds}(Y)=4d(Y)$ (equivalently $\mathfrak{ds}(Y)=d(Y)/2$)?

(Q5.4) If a Brieskorn homology 3-sphere has finite $E_8$-genus, is $g_8(Y)=\overline{g_8}(Y)$, i.e. is the $E_8$-multiplicity unique?

(Q5.5, arXiv Problem 5.1) Do the (11/8)-conjecture-linked bounding-genus inequalities hold for the four explicit infinite Brieskorn families — the even-parameter subfamilies of the four families with $\mathfrak{ds}=n$ listed above, i.e. those with $\mathfrak{ds}=2n$: $|\Sigma(8n-2,8n-1,16n-3)|\ge 3n$, $|\Sigma(8n-1,8n,16n-1)|\ge 3n$, $|\Sigma(8n-2,8n-1,32n^2-8n+1)|\ge 3n$, and $|\Sigma(8n-1,8n,32n^2-1)|\ge 3n$? (Here $|\cdot|$ is the bounding genus; the failure of any one inequality would refute Matsumoto's (11/8)-conjecture.)

(Q5.6, arXiv Question 5.5) Do these same Brieskorn spheres admit $E_8$-boundings with $g_8=2n$?

(Q5.7, arXiv Question 5.6) For the classes $a_k=k[f]-[s]\in H_2(E(n))$ (fiber class minus section class in the elliptic surface $E(n)$): is there an a priori upper bound on $k$ for which $a_k$ can be represented by a smoothly embedded $S^2$? (Known: $k\le 12$ and $k=14$ in $E(1)$; $k\le 42$ in $E(2)$.)

(Q5.8, arXiv Question 5.7) For homology 3-spheres $X_1,X_2$ with $\mathfrak{ds}(X_i)<\infty$ and $\tilde{\mathfrak{ds}}(Y):=\epsilon(Y)\mathfrak{ds}(Y)$, characterize when the signed additivity $\tilde{\mathfrak{ds}}(X_1)+\tilde{\mathfrak{ds}}(X_2)=\tilde{\mathfrak{ds}}(X_1\#X_2)$ holds (the paper records the nontrivial example $X_1=\Sigma(2,3,17)$, $X_2=\Sigma(2,3,13)\#(-\Sigma(2,3,17))$ and expects additivity to be rare).

(Problem 5.1, arXiv Problem 5.2) Find more general constructions producing positive or negative $E_8$-boundings for many homology 3-spheres.

(Q3.1) In the plumbing 4-manifold $S_k$ with intersection form $-E_8\oplus H$ and boundary $Y^-_{2k+1}=\Sigma(2,3,12k+5)$ (where $Y^-_n:=\Sigma(2,3,6n-1)$): can a separating homology 3-sphere $Y\subset S_k$ that splits the form as $-E_8\oplus H$ (i.e. the $H$-summand piece $X$ with $\partial X=Y$) be taken so as to bound an acyclic 4-manifold — that is, can such a $Y$ be taken with trivial class in the homology cobordism group $\Theta^3_{\mathbb{Z}}$ (the abelian group of integral homology 3-spheres modulo homology cobordism, i.e. modulo together bounding a homology product $Y\times[0,1]$)? The source notes this question is equivalent to $\mathfrak{ds}(Y^-_{2k+1})=1$ for any $k$.

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