Schedule for: 22w5045 - Cobordisms, Strings, and Thom Spectra

The cohomology and homotopy groups of moduli spaces of smooth manifolds are some of the most basic invariants of manifold bundles: the former contain all the characteristic classes and the latter classify smooth bundles over spheres. Complete calculations of these groups are challenging, even for the simples compact manifolds. It is then desirable to know, at least, some qualitative information, for example whether these groups are (degreewise) finitely generated. In this talk, I will discuss a method to attack this question which leads to the following theorem: if $M$ is a closed smooth manifold of even dimension $> 5$ with finite fundamental group, then the cohomology and higher homotopy groups of $B\mathrm{Diff}(M)$ are finitely generated abelian groups. This is joint work with M. Krannich and A. Kupers.

The Surgery Exact Sequence provides computable obstructions for deciding the existence and uniqueness of manifold structures. There are many different versions of the surgery exact sequence. One of the most computational versions arises from Ranicki’s interpretation of the obstruction map as the passage from local Poincare ́ duality to global Poincare ́ duality. This interpretation involved the use of additive categories of chain complexes parametrized by a finite simplicial complex $K$ with chain duality. This notion of chain duality is crucial for the whole theory, but it was never proven in the original references. In this talk, I will present recent work with Jim Davis where we provide a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.

We consider the \(E_2\)-algebra $\Lambda M_{∗,1} := \coprod_{g\geqslant0} \Lambda M_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\Omega B\Lambda M_{∗,1}$: it is the product of $\Omega^\infty MTSO(2) $ with the free $\Omega^\infty$-space over a certain space $X$. This extends the classical result $\Omega BM_{∗,1}=\Omega^\infty MTSO(2)$, due to Madsen and Weiss, to the setting of surface bundles parametrised over $S^1$. I will define the space $X$ in the statement and give a brief sketch of the proof, which combines two inputs: $\bullet$ on the one hand, we obtain a structure result for centralisers of mapping classes in generic mapping class groups $\Gamma_{g,n}$, for $g\geqslant0$ and $n\geqslant1$: this uses standard techniques of the theory of mapping class groups, such as arc complexes; $\bullet$ on the other hand, we generalise the theory of operads with homological stability, developed recently by Tillmann et al., to the setting of coloured operads, and compute the group completion of certain "relatively free" algebras over such operads (with respect to a suboperad given by a family of groups); the main application involves a coloured version of Tillmann's surface operad. This is joint work with Florian Kranhold and Jens Reinhold.

Let $E_U$ and \(E_O\) denote the truncated Picard spectra \(pic(KU)[0,3]\) and \(pic(KO)[0,2]\). The main theorem of this work is a computation of the \(k\)-invariants of \(E_U\) and \(E_O\). This computation has several interesting consequences. First, for a nice enough space \(X\), it follows that the cohomology group \(E^0_U(X)\) (respectively \(E^0_O(X)\)) is isomorphic to the Brauer group of complex (resp. real) \(\mathbb{Z}/2\)-graded continuous trace \(C^∗\)-algebras with spectrum \(X\); and isomorphism classes of complex (resp. real) super 2-lines on \(X\). This is essentially a \(\mathbb{Z}/2\)-graded manifestation of the twists of \(K\)-theory arising in classical Dixmier-Douady theory. In particular, if \(X\) is connected then the group of complex (resp. real) super 2-lines on \(X\) is isomorphic to the group of \(ku[0,2]\)-lines (resp. \(ko[0,1]\)-lines) on \(X\). If \(E^c_U\) and \(E^c_O\) denote connective covers of those spectra then it also follows that \(\Omega^{\infty}E^c_U\) and \(\Omega^{\infty}E^c_O\) are equivalent, as infinite loop spaces, to the respective fibers of the covers \(BString \to BSO\) and \(BSpin \to BO\), making it possible to twist String and Spin structures by \(ku[0,2]\) and \(ko[0,1]\)-lines respectively. It is also immediate that all of the above-mentioned spectra appear in different guises in various works of Freed, Hopkins and Teleman. Indeed, the computation of the second \(k\)-invariants of \(E_U\) and \(E_O\) proceeds by giving concrete models of the Picard groupoids of complex and real super lines. Finally, it also follows from this computation that \(E_U\) is abstractly equivalent to the 3-fold suspension of the -3-cotruncation of the Anderson dual of the sphere, i.e. \(\Sigma^3(I_{\mathbb{Z}}[−3,0])\).

Certain mechanisms of spontaneous symmetry breaking in field theories are captured mathematically by Smith homomorphisms, which are maps on bordism groups that change both dimension and tangential structures. Understanding Smith homomorphisms as induced by maps of spectra allows one to compute obstructions to these physical mechanisms and thus constrain the lower-energy behavior of field theories. We apply this perspective to study anomalies free field theories and other examples, as well as elucidate how Smith homomorphisms factor through the crystalline equivalence principle, which relates phases with spatial and internal symmetry groups.

In this talk I will show the complete calculation of the equivariant oriented and unitary bordism groups of surfaces. Of particular interest is the existence of torsion classes coming from surfaces with free action that do not equivariantly bound.

Orbifold TQFTs and gerbes over orbifolds are intimately related by twistings that are completely analogous to the ones appearing in K-theory. In this talk I will revisit gerbes over orbifolds and more general smooth stacks and notice some implications that their structure has for orbifold TQFTs that can be thought of as generalized Thom isomorphisms. Partially joint work with Gonzalez-Segovia-Uribe, and also with Becerra.

Some years ago I posed the following conjecture: Let $f\colon M\to N$ be a map of degree 1 of closed smooth manifolds. Then $\mathrm{crit}(M)\geqslant\mathrm{crit}(N)$. Here $\mathrm{crit}(M)$ (resp. $\mathrm{crit}(N)$) is a minimal number of critical points of $M$ (resp. $N$). In the talk we discuss some results and approaches to the conjecture.

A long-standing open question about mapping class groups of surfaces is whether they are linear, i.e. act faithfully on finite-dimensional vector spaces. In genus zero, for the braid groups, the answer is yes, as proven by Bigelow and Krammer using one of the family of Lawrence representations of the braid groups. Motivated by this, I will describe joint work with Christian Blanchet and Awais Shaukat in which we construct analogues of the Lawrence representations for higher-genus surfaces. A qualitative difference from the genus-zero setting is that our ground ring is non-commutative -- the group ring of the discrete Heisenberg group -- which enriches the representations but has the side effect that they are twisted. One way to untwist them involves the Schrödinger representation and the Stone-von Neumann theorem.

For a finite group \(G\), we define the free $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an isomorphism of its fundamental group onto the direct sum $\mathbb{Z}\oplus H_2(G)$, where $H_2(G)$ is the integral $2$-homology group, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product $G/[G,G]\times S^1\times X^G$, where $\pi_1(X^G)=H_2(G)$. Finally, we present some results about the classification of $G$-topological quantum field theories in dimension two.

In this talk I will explain the construction due to Campbell and Zakharevich and its application that will allow us to speak about the "$K$-theory of manifolds" spectrum. The \(K_0\) of the constructed spectrum recovers the (almost) classical $SK$-groups introduced by Kreck, Karras, Neumann and Ossa. I will explain how to relate the spectrum to the algebraic $K$-theory of integers, and how this leads to certain classical invariants of manifolds when restricted to the lower homotopy groups.

Scissor’s congruence is a classical setup in mathematics that featured in one of Hilbert’s problems in 1900. It asks whether two polytopes can be obtained from one another through a process of cutting and pasting. In the 1970s this question was posed instead for smooth manifolds: which manifolds M and N can be related to one another by cutting M into pieces and gluing them back together to get N? Manifold cut and paste invariants describe when this is possible. In recent work with Mona Merling, Laura Murray, Carmen Rovi and Julia Semikina, we upgraded the group of cut and paste invariants of manifolds to an algebraic K-theory spectrum and lifted the Euler characteristic to a map of spectra. I will discuss how cut and paste invariants relate to cobordism of manifolds and how the novel construction categorifies these invariants. I will also discuss new results on the categorification of cobordism cut-and-paste invariants: the group of invariants preserved by both cobordism and cut and paste equivalence.

The $R$-module Thom spectrum functor was defined by Ando, Blumberg, Gepner, Hopkins, and Rezk in the context of orientations on the $R$-module Thom spectrum. We view the construction from the point of view of structured ring spectra showing that the functor has good monoidal properties. These results allow us to construct $A_\infty$ $R$-algebra structures on various quotients, and compute their Topological Hochschild homology.

After his seminal work on the algebraic $K$-theory of topological spaces, Waldhausen initiated a program to compute the algebraic $K$-theory of the sphere spectrum which contains valuable information regarding the $h$-cobordism spaces of manifolds. In this work, we contribute to this program by studying the algebraic $K$-theory of ring spectra via a root adjunction formalism. In applications of our results, we obtain non-trivial splittings on the $A$-theory spectra of spheres of odd dimensions which provides interesting results regarding the stabilized $h$-cobordism spaces of spheres. Joint work with Christian Ausoni, Tasos Moulinos and Yajit Jain.

Topological field theories (TFTs) provide invariants of smooth manifolds. However what precisely these invariants measure and which manifolds can be distinguished by TFTs remains largely an open problem. In this talk we will report on recent joint work with David Reutter on a class of topological field theories (the "generalized Dijkgraaf-Witten theories") obtained by doing "finite path integration" of certain invertible topological field theories. We obtain positive results: Manifolds satisfying a certain finiteness condition are indistinguishable by generalized Dijkgraaf-Witten theories if and only if they are stably diffeomorphic. This includes 4-manifolds with finite fundamental group and 6-manifolds with finite $\pi_1$ and $\pi_2$, and in some cases leads to examples of topological field theories that distinguish exotic smooth structures.

The third string bordism group is known to be \(\mathbb{Z}/24\mathbb{Z}\). Using the notion of geometric string structure introduced by Waldorf, Bunke-Naumann and Redden have exhibited integral formulas involving the Chern-​Weil form representative of the first Pontryagin class and the canonical 3-​form of a geometric string structure that realize the isomorphism \(\mathrm{Bord}^3_{\mathrm{String}}\cong \mathbb{Z}/24\mathbb{Z}\) (these formulas have been recently rediscovered by Gaiotto-​-Johnson-Freyd--Witten). In the talk I will show how these formulas naturally emerge when one considers the $U(1)$-​valued 3d TQFTs associated with the classifying stacks of Spin bundles with connection and of String bundles with geometric structure. This is based on joint work with Domenico Fiorenza (in preparation).

For a finite group, The equivariant bordism ring is a module over the usual (non-equivariant) cobordism ring. For \(G\) abelian or metacyclic, the equivariant unitary bordism ring is a free module with generators in even degrees. It was conjecture that this should be true for general group, and then B. Uribe promoted the problem at ICM 2018, where he called it the “evenness conjecture for equivariant bordism”. We showed that the conjecture is false by finding explicit counterexamples and explicitly described the 2-dimensional equivariant unitary and oriented bordism groups for all finite groups. This talk is based on joint work with Eric Samperton, Carlos Segovia and Bernardo Uribe.

Free loop spaces arise in many areas of geometry and topology. Simply put, the free loops on a space $X$ is the space of maps from the circle into $X$. This is a main object of study in string topology and has important connections to geodesics on manifolds. In this talk, we discuss a new approach to computing the homology of free loop spaces via topological coHochschild homology, which is an invariant of coalgebras arising from homotopy theory techniques. This approach produces a spectral sequence for the homology of free loop spaces that has an algebraic structure allowing for new computations. This is joint work with Teena Gerhardt and Brooke Shipley.

Many important topological Thom spectra like \(MU\) can be obtained as the realization of their motivic counterparts if the base field is $\mathbb{C}$. In this talk an explicit fibrant resolution of $MU$ with finite coefficients will be presented. It is computed by algebraic varieties only and uses computational miracles of Voevodsky’s framed correspondences (a distant algebraic relative of framed cobordisms) and framed motives in the sense of the speaker and Panin. This is a joint work with Alexander Neshitov.

The simplicial volume and the Euler characteristic are two homotopy invariants of closed (oriented) manifolds which generally have very different properties. A well-known open question of Gromov asks whether the vanishing of the simplicial volume of an aspherical manifold implies the vanishing of its Euler characteristic. In this talk, I will review some facts about the simplicial volume, its connection with bounded cohomology, and also examine its properties from the viewpoint of the cobordism category. Then I will discuss some approaches to relate these two fundamental invariants and present some recent partial results in connection with Gromov's question. This is based on joint work with C. Löh and M. Moraschini.