arXiv research

Global group laws connect equivariant bordism rings to formal group laws.

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…

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, in…

By results of Loeffler and Comezana, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G = S^1 x ... x S^1, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of D. Sinha. …

We study the eta invariants of compact flat spin manifolds of dimension n with holonomy group cyclic of odd prime order p. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when p=n=3. We use the expressions ob…

Invariants for colored links found, with topological protection of certain knots.

We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the …

We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods.

A modular tensor category $\mathcal{C}$ gives rise to a Reshetikhin-Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded $\mathcal{C}$-coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. …

Let $G$ be a compact connected Lie group acting on a stable complex manifold $M$ with equivariant vector bundle $E$. Besides, suppose $φ$ is an equivariant map from $M$ to the Lie algebra $\mathfrak{g}$. We can define some equivalence relation on the triples $(M, E, φ)$ such that the set of equivalence classes form an …

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by …

We show an equivariant bordism principle for constructing metrics of positive scalar curvature that are invariant under a given group action. Furthermore, we develop a new codimension-2 surgery technique which removes singular strata from fixed point free $S^1$-manifolds while preserving equivariant positive scalar cur…

We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging $G$-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a …

We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples…

New perspective on APS indices preserves orientations and gradings through bordisms.

We call a Morse function $f$ on a closed manifold $k$-constrained if neither $f$ nor $-f$ has critical points of indefinite Morse index $< k$. In this paper we study bordism groups of $k$-constrained Morse functions, and thus interpolate between the case $k = 1$ of bordism groups of Morse functions (computed by Ikegami…

This note provides a computation of the bordism groups of K-Witt spaces for fields K with characteristic 2. We provide a complete computation for the unoriented bordism groups. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambi…

The group of bordism classes of unoriented surfaces in 4-space is determined. The bordism classes are characterized by normal Euler numbers,double linking numbers, and triple linking numbers.

Study PL bordism theories with quantitative bounds on filling simplices.

Quantum field theory uses Lorentzian bordisms to describe time evolution.

We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes $ζ_n$ take the places of the prime numbers, and this allows us to di…

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…

We define new bordism and spin bordism invariants of certain subgroups of the mapping class group of a surface. In particular, they are invariants of the Johnson filtration of the mapping class group. The second and third terms of this filtration are the well-known Torelli group and Johnson subgroup, respectively. We i…

The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and t…

Generalised characteristic classes are constructed for bordism cohomologies which allow a natural extension of classical genera to these bordism cohomology rings taking values in singular cohomology.

Model for assembly map of bordism-invariant functors.

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metr…

We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We show that unitary representations of $G$ produce functors from the category of …

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…

Let $Γ$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $\mathrm{B} Γ$. The lower bounds are formulated in terms of the part …

The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.

Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.

We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…

Example of group action on surface that can't extend to 3-manifold.

We prove a folklore result which gives a sequence of necessary and sufficient conditions for a stably symplectic manifold to define a nilpotent element in the symplectic bordism ring.

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for high-dimensional compact ANR homology manifolds is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

Paper establishes equivalence between algebraic and functorial QFTs.

Study shows groups can act on torus without extending to 3-manifold.

Proves infinite bordism groups for certain manifolds with positive scalar curvature.

The goal of this paper is to study the Pontrjagin dual of (reduced) 4-dimensional Spin bordism. That is to say, we consider the functor from the category of topological spaces to the category of compact abelian groups that associates to each space X the compact group of homomorphisms from the reduced 4-dimensional Spin…

This note corrects an error in the char(K)=2 case of the author's computation of the bordism groups of K-Witt spaces for the field K.

Ricci-positive manifolds span the kernel of the $\hat A$-genus in rational Spin bordism.

In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examin…

A modular functor is constructed from non-semisimple 3d TFTs.

Abstract: Mapping 3-manifold bordisms to topological orders and domain walls.

We give bordism-finiteness results for manifolds with semi-simple group action. Consider the class of oriented manifolds which admit a circle action with isolated fixed points such that the action extends to an $S^3$-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler c…

Let $(Y,g)$ be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of $Y$ also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result wit…

New invariants detect corks obstructing homology ball extensions.