arXiv research

New polynomials detect non-rotatable knotoid shapes.

In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the…

Uniformly proves index invariance for signature operators on manifolds.

Paper proves existence of Dirac-harmonic maps with trivial index.

The thesis defines and proves invariants for manifolds of bounded geometry.

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with c…

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable spa…

We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.

We provide an interpretation of the APS index theorem of Piazza-Schick and Zeidler in terms of coarse homotopy theory. On the one hand we propose a motivic version of the boundary value problem, the index theorem, and the associated secondary invariants. On the other hand, we discuss in detail how the abstract version …

It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor's invariants with repeated indices are invariants not only of isotopy, but also of self C_k-moves. A self C_k-move is a natural generalization of link homotopy based on certain degree k clasper su…

A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.

We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators th…

A compact polyhedron $X$ is said to have the Bounded Index Property for Homotopy Equivalence (BIPHE) if there is a finite bound $\mathcal{B}$ such that for any homotopy equivalence $f:X\rightarrow X$ and any fixed point class $\mathbf{F}$ of $f$, the index $|\mathrm{ind}(f,\mathbf{F})|\leq \mathcal{B}$. In this note, w…

Paper extends BIP to nilmanifold products and characterizes fixed points.

This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.

We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $Spin^c$-Dirac operators. As an application we give new examples of non-trivial Ham…

Study finds bound on energy of minimal spheres on complex manifolds.

Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.

This article surveys the relations among local and nonlocal invariants in Atiyah-Singer index theory. We discuss the local invariants that arise from the heat equation approach to the index theorem for geometric operators, as well as the nonlocal invariants (the eta invariant, the determinant of the Laplacian/analytic …

Defines discrete differential geometry concepts in homotopy type theory.

This is a sequel to the paper "The signature package on Witt spaces, I. Index classes" by the same authors. In the first part we investigated, via a parametrix construction, the regularity properties of the signature operator on a stratified Witt pseudomanifold, proving, in particular, that one can define a K-homology …

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singu…

In this paper, we study Conley theory in Hilbert spaces and make some refinement of the construction of the stable Conley index developed by Gȩba, Izydorek, and Pruszko. For instance, we allow subspaces other than invariant subspaces in the the construction. As a main result, we show that the resulting stable Conley in…

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the Gromov-Lawson Conjecture for such manifolds in the case of particular type of singularitie…

Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.

Study of psc metrics via block diffeomorphisms and cubical sets.

Given a family of Dirac operators with vanishing spectral flow we construct a thin-invariant rank-one field theory in the sense of Turner and Willerton arXiv:math.AT/0201116. Our construction of the field theory generalizes the one of the index gerbe by Lott, arXiv:math.DG/0106177, and it also complements the relation …

Let $M$ be a manifold homotopy equivalent to the complex projective space $\C P^m$. Petrie conjectured that $M$ has standard total Pontrjagin class if $M$ admits a non-trivial action by $S^1$. We prove the conjecture for $m<12$ under the assumption that the action extends to a nice $Pin(2)$-action with fixed point. The…

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maxim…

The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…

Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.

We give a construction to remove coincidence points of continuous maps on graphs (1-complexes) by changing the maps by homotopies. When the codomain is not homeomorphic to the circle, we show that any pair of maps can be changed by homotopies to be coincidence free. This means that there can be no nontrivial coincidenc…

New index formula connects numerical and $K$-theoretic indices.

We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct the associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of the homotopy groups of Diff(Z).

We study noncommutative eta- and rho-forms for homotopy equivalences. We prove a product formula for them and show that the rho-forms are well-defined on the structure set. We also define an index theoretic map from L-theory to C*-algebraic K-theory and show that it is compatible with the rho-forms. Our approach, which…

Study shows surfaces without certain curves have infinite orbit graph.

We prove exponential growth rate of contractible closed geodesics for an arbitrary bumpy metric on manifolds of the form X#Y, where the fundamental group of X has a subgroup of finite index at least 3 and Y is simply connected and not a homotopy sphere.

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion of index for vector fields, the $q=n$ case. The existence of a continuous map $g…

We show that the disk complex of a genus $g>1$ Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of $(2g-2)$-dimensional spheres. This implies that genus $g>1$ Heegaard surfaces for the 3-sphere are topologically minimal with index $2g-1$.

In the framework of fibred cusp operators on a manifold $X$ associated to a boundary fibration $Φ: \pa X\to Y$, the homotopy groups of the space of invertible smoothing perturbations of the identity are computed in terms of the K-theory of $T^{*}Y$. It is shown that there is a periodicity, namely the odd and the even h…

Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective ma…

Classifies crossings in tangles on surfaces, finding no nontrivial indices.

In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive scalar curvature)? More generally: what is the topology of this space? For example, …

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…

Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of π_2(L) in π_2(N) has finite index. We make no assumption on the Maslov class of L, and we make no orientability assumptions. The homotopy assumption is e…

Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $ρ$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theo…