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Register a new userPin(2)-equivariant KO-theory and intersection forms of spin four-manifolds
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Jianfeng Lin
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Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$-spheres. As an application, we give explicit constrains on the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres $pmSigma(2,3,6kpm1)$. Along the way, we also give an alternative proof of Furuta-Kametannis improvement of 10/8-theorem for closed spin-4 manifolds.
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In studying the 11/8-Conjecture on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a 10/8+4-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch spectral sequence.
For a 3-manifold $M$ with $b_1(M)=1$ fibered over $S^1$ and the fiberwise gradient $xi$ of a fiberwise Morse function on $M$, we introduce the notion of amidakuji-like path (AL-path) on $M$. An AL-path is a piecewise smooth path on $M$ consisting of edges each of which is either a part of a critical locus of $xi$ or a flow line of $-xi$. Counting closed AL-paths with signs gives the Lefschetz zeta function of $M$. The moduli space of AL-paths on $M$ gives explicitly Lescops equivariant propagator, which can be used to define $mathbb{Z}$-equivariant version of Chern--Simons perturbation theory for $M$.
Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL embedded closed surface. One can arrange the embedding to have at most one non-locally-flat point, and near the point the topology of the embedding is encoded in the singularity knot $K$. If $K$ is slice, then $W$ has a smooth spine, i.e., deformation retracts onto a smoothly embedded surface. Using the obstructions from the Heegaard Floer homology and the high-dimensional surgery theory, we show that $W$ has no smooth spines if $K$ is a knot with nonzero Arf invariant, a nontrivial L-space knot, the connected sum of nontrivial L-space knots, or an alternating knot of signature $<-4$. We also discuss examples where the interior of $W$ is negatively curved.
We provide three 3-dimensional characterizations of the Z-slice genus of a knot, the minimum genus of a locally-flat surface in 4-space cobounding the knot whose complement has cyclic fundamental group: in terms of balanced algebraic unknotting, in terms of Seifert surfaces, and in terms of presentation matrices of the Blanchfield pairing. This result generalizes to a knot in an integer homology 3-sphere and surfaces in certain simply connected signature zero 4-manifolds cobounding this homology sphere. Using the Blanchfield characterization, we obtain effective lower bounds for the Z-slice genus from the linking pairing of the double branched cover of the knot. In contrast, we show that for odd primes p, the linking pairing on the first homology of the p-fold branched cover is determined up to isometry by the action of the deck transformation group on said first homology.
Given a closed four-manifold $X$ with an indefinite intersection form, we consider smoothly embedded surfaces in $X setminus $int$(B^4)$, with boundary a knot $K subset S^3$. We give several methods to bound the genus of such surfaces in a fixed homology class. Our techniques include adjunction inequalities and the $10/8 + 4$ theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold and show that the set of H-slice knots can detect exotic smooth structures on closed $4$-manifolds.
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