arXiv research

Established a stable cohomotopy refinement for Pin(2) monopole invariants.

The monopole map defines an element in an equivariant stable cohomotopy group refining the Seiberg-Witten invariant. This first of two articles presents the details of the definition of the stable cohomotopy invariant and discusses its relation to the integer valued Seiberg-Witten invariant.

This is a survey article on the stable cohomotopy refinement of Seiberg-Witten invariants containing also new results, for example: - Stable cohomotopy groups describe path components of certain mapping spaces. - Relation of stable cohomotopy invariants to Seiberg-Witten invariants without restriction on Betti numbers.…

The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a connected sum is the smash product (in the sense of equivariant spectra) of the invaria…

The paper characterizes stable cohomotopy groups in codimensions two and three, linking algebraic and geometric perspectives.

Study cohomotopy classes for 4-manifolds using complex spin structures.

S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable loc…

New mathematical framework connects M-theory charges to stable homotopy groups.

We will define a version of Seiberg-Witten-Floer stable homotopy types for a closed, oriented 3-manifold $Y$ with $b_1(Y) > 0$ and a spin-c structure $\mathfrak{c}$ on $Y$ with $c_1(\mathfrak{c})$ torsion under an assumption on $Y$. Using the Seiberg-Witten-Floer stable homotopy type, we will construct a gluing formula…

Differential Cohomotopy theory predicts brane interactions via chord diagrams.

Geometrically computes cohomotopy groups for higher-dimensional manifolds.

We show that every positive definite closed 4-manifold with $b_2^+>1$ and without 1-handles has a vanishing stable cohomotopy Seiberg-Witten invariant, and thus admits no symplectic structure. We also show that every closed oriented 4-manifold with $b_2^+\not\equiv 1$ and $b_2^-\not\equiv 1\pmod{4}$ and without 1-handl…

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a …

New framework solves string theory's RR-field tadpole cancellation problems.

Proves a formula for a special invariant of 4-manifolds.

O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these…

Diagonalizes tautological classes of definite 4-manifolds.

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…

In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of eq…

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

Study cohomotopy sets of simply connected 7-manifolds using suspension decompositions.

Cohomotopy charge quantization implies String structure on M5-branes.

The paper characterizes cohomotopy sets of specific manifolds.

In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in the earlier …

We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estima…

In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting. Instead of looking at the solutions, if we can regard some kind of homotopy class…

The paper determines modular cohomotopy groups up to extensions using classical and unstable homotopy methods.

Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which hav…

Resolves anomaly cancellation for M5-brane in M-theory.

Refines quantum annular homology using stable homotopy methods.

In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …

We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positi…

We consider the hypothesis that the C-field 4-flux and 7-flux forms in M-theory are in the image of the non-abelian Chern character map from the non-abelian generalized cohomology theory called J-twisted Cohomotopy theory. We prove for M2-brane backgrounds in M-theory on 8-manifolds that such charge quantization of the…

Study characterizes cohomology and homotopy types for M-theory extensions.

In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…

New stable homotopy refinement of quantum annular Khovanov homology.

We review the construction and context of a stable homotopy refinement of Khovanov homology.

We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that ou…

New manifold construction yields Baire-1 functions as cohomotopy groups.

New homotopy refinements for tangle invariants.

Refines Khovanov homology using signed Burnside categories.

Geometrically counts embedded circles and sections of vector bundles.

Study Kähler-Einstein potentials on stable varieties near singularities

Higher gauge theory via differential nonabelian cohomology

Develops theory for stable capillary minimal hypersurfaces in half-space.

We define stable homotopy refinements of Khovanov's arc algebras and tangle invariants.

We develop a Chern character map for twisted equivariant non-abelian cohomology.

Paper constructs Chern character for higher twists and shows isomorphism between K-theory and cohomology.