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Register a new userIntersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant
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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.
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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.
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.
We generalize the Mahowald invariant to the $mathbb{R}$-motivic and $C_2$-equivariant settings. For all $i>0$ with $i equiv 2,3 mod 4$, we show that the $mathbb{R}$-motivic Mahowald invariant of $(2+rho eta)^i in pi_{0,0}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Adams classical $v_1$-periodic families, and for all $i > 0$, we show that the $mathbb{R}$-motivic Mahowald invariant of $eta^i in pi_{i,i}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Andrews $mathbb{C}$-motivic $w_1$-periodic families. We prove analogous results about the $C_2$-equivariant Mahowald invariants of $(2+rho eta)^i in pi_{0,0}^{C_2}(S^{0,0})$ and $eta^i in pi_{i,i}^{C_2}(S^{0,0})$ by leveraging connections between the classical, motivic, and equivariant stable homotopy categories. The infinite families we construct are some of the first periodic families of their kind studied in the $mathbb{R}$-motivic and $C_2$-equivariant settings.
We analyze two different fibrations of a link complement M constructed by McMullen-Taubes, and studied further by Vidussi. These examples lead to inequivalent symplectic forms on a 4-manifold X = S x M, which can be distinguished by the dimension of the primitive cohomologies of differential forms. We provide a general algorithm for computing the monodromies of the fibrations explicitly, which are needed to determine the primitive cohomologies. We also investigate a similar phenomenon coming from fibrations of a class of graph links, whose primitive cohomology provides information about the fibration structure.
We prove that the 2-primary $pi_{61}$ is zero. As a consequence, the Kervaire invariant element $theta_5$ is contained in the strictly defined 4-fold Toda bracket $langle 2, theta_4, theta_4, 2rangle$. Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.
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