# Manifolds, homotopy theory, and related topics

General information

Time & place: Biweekly Tuesdays at 15:00 UK time (10:00 East Coast, 16:00 Germany) on Zoom.

Format: Talks are 60min and will not be recorded.

Mailing list: You can sign up to the email list using this form.

Meeting-ID: Will be distributed to those who subscribed to the mailing list.

Organisers: Manuel Krannich, Alexander Kupers, and Oscar Randal-Williams

Schedule in 2024

The Torelli group of a surface is the kernel of the action of the mapping class group on the first homology of the surface. It is a longstanding open problem to determine whether or not the Torelli group is finitely presented for closed oriented surfaces of genus at least 3. We will discuss some recent work of the author that rules out the simplest obstruction to the Torelli group being finitely presented. In particular, we will show that the second rational homology of the Torelli group is finite dimensional for all surfaces of sufficiently large genus.

Kirby-Siebenmann showed that, given a closed manifold M, there are only finitely many smooth manifolds, up to diffeomorphism, which are homeomorphic to M. This is not true equivariantly. Using computations of Ewing, Schultz has shown that, for certain primes p and sufficiently large n, S^{2n} can be given a smooth Z/p-action with infinitely many Z/p-smoothings. I will explain this example and give a generalization to other manifolds.

## Mar 26th Marco Volpe

Traces of dualizable categories and functoriality of the Becker-Gottlieb transfers

For any fiber bundle with compact smooth manifold fiber X ⟶ Y, Becker and Gottlieb have defined a "wrong way" map S[Y] ⟶ S[X] at the level of homology with coefficients in the sphere spectrum. Later on, these wrong way maps have been defined more generally for continuous functions whose homotopy fibers are finitely dominated, and have been since referred to as the Becker-Gottlieb transfers. It has been a long standing open question whether these transfers behave well under composition, i.e. if they can be used to equip homology with a contravariant functoriality.

In this talk, we will approach the transfers from the perspective of sheaf theory. We will recall the notion of a locally contractible geometric morphism, and then define a Becker-Gottlieb transfer associated to any proper, locally contractible map between locally contractible and locally compact Hausdorff spaces. We will then use techniques coming from recent work of Efimov on localizing invariants and dualizable stable infinity-categories to construct fully functorial "categorified transfers". Functoriality of the Becker-Gottlieb transfers is then obtained by applying topological Hochschild homology to the categorified transfers.

If time permits, we will also explain how one can use similar methods to extend the Dwyer-Weiss-Williams index theorem for compact topological manifolds fiber bundles to proper locally contractible maps. In particular, this shows that the homotopy fibers of a proper locally contractible map are homotopy equivalent to finite CW-complexes. Therefore, it is still unclear whether functoriality of the transfers associated to maps with finitely dominated homotopy fibers holds.

This is a joint work with Maxime Ramzi and Sebastian Wolf.

We recently described a new geometric model for the classifying space of diffeomorphisms of 3-dimensional handlebodies as a specific open locus of the moduli space of Riemann surfaces. In particular combining our model with the work of Chan-Galatius-Payne on the hairy graph complex and on the tropical moduli space of curves we obtain a large number of unstable homology classes for this classifying space.

I will explain the main ideas that go toward proving our results, and depending on the remaining time I will also explain at the end how our model allows us to state a conjectural generalization of the isomorphism of Chan-Galatius-Payne to non-trivial local system coefficients.

This work is joint with Dan Petersen.

Published in 1986, Quinn provided a proof that for a simply-connected 4-manifold, any homeomorphism pseudo-isotopic to the identity is actually isotopic to the identity. Additionally for smooth manifolds, diffeomorphisms that are pseudo-isotopic are smoothly stably isotopic. Both of these results are fundamental to the world of 4-manifolds. Part of the strategy Quinn's employs for deriving these results revolves reducing the problem down to understanding and manipulating a specific collection of embedded disks. There is one particular "move" that is critical to Quinn's argument for both the smooth stably isotopic and topological pseudo-isotopy implies isotopy: the replacement criterion. However, the justification for using this move is incorrect. I plan to discuss the replacement criterion and give a way to circumvent it in order to complete Quinn's proofs.

In this paper we show an additivity result for equivariant cobordism categories, à la Steimle. Through the scanning map, it corresponds to the classical isotropy separation sequence of for genuine G-spectra. This first recovers the computation of the homotopy type of the equivariant cobordism category by Galatius/Szucs. From the proof, we also derive a fiber sequence of moduli spaces that enables us to describe the homology of the BDiff^G(M), for a compact G-manifold M verifying certain properties, in a range of degrees.

The zeroeth complex topological K-theory of a space is highly computable, and encodes complex vector bundles up to stabilization. However, bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest finite CW complexes.

I will discuss joint work with Hood Chatham and Yang Hu to address this problem. Our main result is to compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. Our methods combine Weiss calculus with computational techniques from chromatic homotopy theory. I will explain why the particular chromatic theories that we use – higher real K-theories – are particularly appropriate for this problem, and I will survey some of the computational tools at our disposal. I will also discuss some future directions and indicate where challenges arise in extending our methods.

Unstable algebraic K-theory: E-infinity structure and the rank filtration

In previous work (joint with Dustin Clausen), we introduced the reductive Borel-Serre categories: for a given ring A and any finitely generated projective module M, we associate a 1-category RBS(M) whose geometric realisation functions as a model for unstable algebraic K-theory. The RBS(M)'s assemble to a monoidal category M(A) whose realisation group completes to the K-theory space K(A), giving a sense in which the RBS(M)'s stabilise to K(A).

We have been curious about exploring other questions of stability, e.g. homological stability of the RBS(M)'s. Unfortunately, we have been unable to plug into any standard tools as these generally require some kind of homotopy commutativity and M(A) is only monoidal, not symmetric, not even braided, so its realisation is a priori only an E1-space.

In recent work, we overcome this difficulty by showing that |M(A)| naturally admits an E-infinity structure! This completely opens up the investigation of stability. In this talk, we will present the main ideas behind this proof and also explore a natural rank filtration of M(A).

Function spaces have been a fundamental object of study for decades across widely varying fields of mathematics. In this talk, I’ll discuss a recipe that describes the space of functions (continuous/algebraic) between two (smooth, projective) algebraic varieties as derived indecomposables of a certain module over a graded commutative monoid in some suitably constructed symmetric monoidal category. Among other things, our result can be interpreted as an (further) algebrization (in the sense of homotopical algebra) of Bendersky-Gitler’s results from 1991, which essentially added up to saying that the space of continuous maps between two nice topological spaces has a ‘configuration space model’.

In joint work with Brown, Chan, and Payne, we describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties, and use it to give lower bounds on the dimensions of these cohomology groups. One step is to construct a coproduct on Quillen's spectral sequence, abutting to the rational homology of the one-fold delooping of the algebraic K-theory space of the integers, making it a spectral sequence of Hopf algebras. We also relate this spectral sequence to one involving Kontsevich's graph complexes.

It is a classical fact of rational homotopy theory that the E_infinity-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still E_n-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of E_n-algebras, closely related to Koszul duality for the E_n-operads. We will also see that these results are essentially sharp: if the R-valued cochains of S^n have square-zero E_{n+1}-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.

No talks between July 17th and mid September.

Schedule in 2023

I will describe a new functor calculus, reminiscent of orthogonal calculus, for functors from FI, the category of finite sets and injections, to a stable presentable infinity-category. As in other functor calculi, homogeneous FI-objects are classified by "Taylor coefficients," and FI-calculus has the agreeable property that a Taylor tower can always be recovered from its coefficients along with canonical morphisms between those coefficients. I will also discuss how the phenomenon of representation stability emerges as a facet of FI-calculus, so that FI-calculus can be understood as the elaboration of representation stability into a functor calculus in the stable infinity-categorical setting.

The homology of the unordered configuration spaces of a graph forms a finitely generated module over the polynomial ring generated by its edges; in particular, each Betti number is eventually equal to a polynomial in the number of particles, an analogue of classical homological stability. We show that the degree of this polynomial is captured by a connectivity invariant of the graph, and we compute its leading coefficient explicitly in terms of cut counts and vertex valences. This "stable" (asymptotic) homology is generated entirely by the fundamental classes of certain tori of geometric origin. We also discuss conjectural analogous phenomena in the ordered setting. This talk represents joint work with Byung Hee An and Gabriel Drummond-Cole.

The Loday-Quillen-Tsygan theorem identifies the cyclic cohomology of an algebra A as the large N limit of Lie algebra cohomology of gl_N(A), i.e., matrices with coefficients in A. When A has a nondegenerate pairing, one can ask about deformations of these constructions ("quantizations") and compatibility with this LQT map. Remarkably, there is a special case that encodes the Gaussian unitary ensemble, and it offers a homological view on the emergence of ribbon graphs in this domain of probability theory. More broadly, this quantum LQT construction offers a model of gauge/string dualities in the topological setting, with potential applications to mirror symmetry. This work is joint with Greg Ginot, Alastair Hamilton, and Mahmoud Zeinalian.

We present a homotopy theoretic method for calculating Ext groups between polynomial functors from the category of (finitely generated, free) groups to abelian groups. It enables us to extend the range of what can be calculated. In particular, we can calculate torsion in the ext groups, about which very little seems to have been known. We will discuss some applications to the stable cohomology of Aut(F_n), based on a theorem of Djament. Joint with Marco Nervo.

Given a nonunital commutative S-algebra I, T(I), the Andre-Quillen homology of I, is a derived version of I/I^2.

We describe how I is built from T(I), and how T(I) is built from I.

On one hand the natural map from I to T(I) is the beginning of a tower under I, with nth fiber built from Com(n) and the n-fold smash product of T(I) with itself. This can be identified with the Goodwillie tower of the identity in commutative S-algebras.

On the other hand this same natural map from I to T(I) is the beginning of an increasing filtration on T(I) with nth cofiber built from Lie(n) and the n-fold smash product of I with itself. This is the filtration used by Behrens and Rezk in their work on chromatic unstable homotopy, and we give and outline of the construction that allows one to check that there is agreement between their construction of this filtration and my earlier one.

As an application, if X is a connective spectrum, with 0th space X(0), the above theory gives interesting spectral sequences for computing the homology of X from the homology of X(0) and vice-versa: one lets I(X) be (roughly) the suspension spectrum of X(0) and computes that T(I(X)) = X.

Kontsevich’s invariants as topological invariants of configuration space bundles

The result of Watanabe(’18) showed that Kontsevich’s invariants can distinguish smooth fiber bundles that are isomorphic as topological fiber bundles. Given a framed smooth fiber bundle E over M (with homology sphere fibers), Kontsevich’s invariants are defined by considering the (Fulton-MacPherson compactified) configuration space bundle of E and doing some intersections in its total space to get an intersection number. Since the Fulton-MacPherson compactification is obtained by doing a sequence of real blow-ups, which depends on the smooth structure in an essential way, yet intersection theoretical invariants usually do not depend on the smooth structure, it is plausible that different smooth structures on E yield different topological structures on the configuration space bundles of E, and Kontsevich’s invariants only depend on the topological structure of the configuration space bundles. We verify that this is indeed the case.

For a finitely dominated map of spaces f:A-->B, the Becker-Gottlieb transfer is a certain "wrong-way'' map f^! from the suspension spectrum of B to that of A. Roughly speaking, this transfer encodes the Euler characteristics of the connected components of the fibers of f. For a composable pair f:A-->B and g:B-->C of finitely dominated maps, the composition gf:A-->C is again finitely dominated, and it is natural to ask whether (gf)^! = f^!g^!. That is, if the transfer is functorial with respect to composition of finitely dominated maps. Several attempts to prove this functoriality appeared in the literature, but it is still open.

In my talk, I will explain the subtlety of this functoriality from several directions. First, I will discuss a wider context in which an analogous transfer can be defined and demonstrate that in this wider generality functoriality fails. I will then give a general formula from a joint work with Cnossen, Ramzi, and Yanovski, for the transfer along the composition gf. Besides f^!g^!, this formula involves the ``traces of monodromy'' along various free loops in B. Based on this formula, I will present several cases in which the transfers do compose and a result of Klein, Malkiewich, and Ramzi on the functoriality of transfers for arbitrary finitely dominated maps on the level of \pi_0.

Many of the functors typically studied using Orthogonal Calculus, such as BO or BTOP, admit a lax symmetric monoidal structure, yet such structures have not played a role in Orthogonal Calculus so far.

I will explain a proof of the fact that the Taylor approximations of a lax symmetric monoidal functor are themselves lax symmetric monoidal, using methods of infinity-category theory such as Day convolution. I will also explain what kind of maps are induced on the derivative spectra of a lax symmetric monoidal functor.

I will talk about joint work with Hebestreit, Weiss, and Winges. First, I will give a brief introduction to homology manifolds from a modern perspective and indicate that the underlying homotopy type of a compact homology manifold is a Poincare duality complex, and hence is equipped with a Spivak normal fibration. It is an old theorem of Ferry and Pedersen that this Spivak normal fibration admits a (canonical) reduction to a stable euclidean bundle, or equivalently that a compact homology manifold admits a (canonical) degree one normal map from a closed topological manifold. Much of the literature on homology manifolds rests on the existence of such a degree one normal map. I will then show that this fact and the seminal results of Bryant-Ferry-Mio-Weinberger on the surgery theory of homology manifolds contradict each other.

In the final part, I will summarize which parts of BFMW should be considered open, and under what assumptions their main result on the surgery classification should hold true, based on communications with Weinberger. Taking this for granted, our approach in fact yields an example of a homology manifold whose Spivak fibration does not admit a lift to a stable euclidean bundle.

It was conjectured by Milnor that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper (arXiv:2303.15347) is a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated.

After a brief crash course in the background material we will discuss the topological and geometric construction of the space. In particular we will see how the mapping class group of S^3xS^3 can be used to build a complete space with twisted actions that are geometrically compatible with Ricci curvature.

Schedule in 2022

Recall that a smooth manifold is said to be almost closed if its boundary is a homotopy sphere. In this talk, I will describe joint work with Robert Burklund that gives a homotopy-theoretic classification of high-dimensional metastably connected almost closed manifolds. (Here I say that an n-manifold is metastably connected if it is approximately (n/3)-connected.) More precisely, we show that such a manifold is determined up to diffeomorphism by its underlying stable homotopy type, its stable normal bundle, and a homotopical quadratic form refining the Poincare duality pairing. In the case of (n-1)-connected 2n-manifolds, our classification recovers that of C.T.C. Wall. When combined with earlier joint work with Robert Burklund and Jeremy Hahn, this may be used to obtain classification results for high-dimensional metastably connected closed manifolds.

In recent work with Ben Antieau and Thomas Nikolaus, we develop new methods to compute K-theory of Z/p^n and related rings, based on prismatic cohomology. This approach can be turned into an algorithm, which we implement. The same methods also allow us to prove that K-theory of Z/p^n vanishes in large enough even degrees, and to give an explicit formula for the orders in large odd degrees. In this talk, I want to give an overview over the ingredients of these computations.

On the rational homology of diffeomorphism groups for some odd-dimensional manifolds

I'll talk about new results on the rational cohomology of the classifying space of Diff_\partial (U_{g,1}^n), where U_{g,1}^n is the connected sum of g copies of S^n \times S^{n+1}, minus a disc, which we consider as an odd-dimensional analogue of W_{g,1}^n, the connected sum of g copies of S^n \times S^n.

We compute the rational cohomology of BDiff_\partial (U_{g,1}^n) for large g and in degrees up to n-4. The answer looks superficially similar to the even-dimensional case in the sense that the cohomology is an exterior algebra in some generalized Miller--Morita--Mumford classes, with some notable differences.

The proof relies on the classical approach to diffeomorphism groups via surgery theory and pseudoisotopy theory. A new ingredient is the result that the gluing map BDiff_\partial (D^{2n+1}) \to BDiff_\partial (U_{g,1}^n) induces the trivial map in rational homology in the concordance stable range. This relies on the work by Botvinnik-Perlmutter on moduli spaces of odd-dimensional manifolds.

This is joint work with Jens Reinhold.

Given a closed manifold M one can construct two Euler characteristics in A-theory, one coming from the Euler class of the tangent bundle and the other one from the fact that M is (homotopy equivalent to) a finite complex. The identification of the two is determined by the underlying simple homotopy type. I will explain that after passing from A-theory to Hochschild homology, this identification can be extracted from configuration spaces of (at most 2) points on M. More concretely, I will discuss a construction that extracts the Dennis trace of Reidemeister/Whitehead-torsion from configuration spaces of points. Along the way we will see how this type of structure controls a relative intersection product which has applications in string topology.

This is joint work with Pavel Safronov.

Low-degree cohomology of arithmetic groups with unitary or finite-dimensional coefficients

We compute the cohomology of semi-simple Lie groups with respect to (possibly infinite-dimensional) unitary representations and in degrees below the rank. Using a new Shapiro isomorphism which is based on a breakthrough result by Leuzinger-Young in geometric group theory we are able to extend the computation from the ambient Lie group to its lattices. This is based on joint work with Uri Bader.

Let R be a ring. The term unstable algebraic K-theory will refer to any (family of) anima K(R,n) built entirely out of linear algebra internal to R^n through which the canonical maps BGL_n(R) → K(R) factorise. A classical example is Quillen's plus-construction BGL_n(R)^+. Ideally, we want a model for unstable algebraic K-theory to be closer to K(R) than BGL_n(R) is in terms of its nature and properties; for example the fundamental group of the plus-construction is closer to K_1(R) than GL_n(R) is. The term unstable algebraic K-theory was used in the 1970's by Dennis and Stein in a survey of the functor K_2, and classically unstable algebraic K-theory has been used to derive many important computational results about ``stable'' algebraic K-theory.

In joint work with Dustin Clausen, we introduce a new model for unstable algebraic K-theory inspired by a detailed study of the so-called reductive Borel--Serre compactification of locally symmetric spaces. In this talk I will introduce this model in detail; I will go through the main results and calculations that we have obtained so far and shed light on the most important aspects of the proofs.

The classifying space of the surface category is equivalent to \Omega^{\infty-1} MTSO_2 by work of Galatius-Madsen-Tillmann-Weiss. I will define a filtration of this by infinite loop spaces where the g-th stage only contains contributions from moduli spaces of surfaces of genus at most g. The associated graded of this filtration can be shown to consist of free infinite loop spaces built from curve complexes.

With rational coefficients this filtered infinite loop space yields a spectral sequence whose E_1 page contains (the dual of) all unstable homology groups of mapping class groups of closed surfaces, and which converges to the spectrum homology of MTSO_2. As a consequence, one can for example conclude that the group H_{14}(B\Gamma_5), which was shown to be non-zero by Chan-Galatius-Payne, has rank at most 2.

After setting up the filtration and explaining the spectral sequence, I will go into more detail about the tools used to identify the associated graded.

## No talks between Aug 31st and Oct 17th.

Labeled configuration spaces B_k(M;X) of a manifold M with labels in a spectrum X generalize the notion of unordered configuration spaces B_k(M)=B_k(M;S^0). Knudsen identified labeled configuration spaces in M with the topological Quillen objects of certain spectral Lie algebras. This allows us to extract information about the mod p homology of B_k(M;X) using a bar spectral sequence and power operations on spectral Lie algebras, following the work of Knudsen on the rational homology of B_k(M;X) and Brantner-Hahn-Knudsen on their Morava E-theory. In this talk, I will explain how to compute the E^2-page of this bar spectral sequence via a May spectral sequence when p=2. Time permitting, I will talk about ongoing work with Andrew Senger on identifying the higher differentials via one-parameter deformation of comonads.

Given two closed h-cobordant manifolds M and M', how different can the homotopy types of the diffeomorphism groups Diff(M) and Diff(M') be? We will see that the homotopy groups of these two spaces are the same “up to extensions” in positive degrees in a range. Contrasting this fact, I will present an example of h-cobordant manifolds with different mapping class groups. In doing so, I will introduce a moduli space of “h-block” bundles and understand its difference with the moduli space of ordinary block bundles.

In a famous paper, Sullivan showed that the rational homotopy theory of finite type nilpotent spaces can be encoded in a fully faithful manner by mapping it to the homotopy category of commutative differential graded algebras over the rational numbers. For integral homotopy theory, a result of Mandell shows that it is faithfully captured by the integral cochains equipped with their E-infinity structute. This functor is however not full. I will explain a way of fixing this problem inspired by work of Toën, using cosimplicial binomial rings instead of E-infty differential graded algebras.

Using Seiberg-Witten theory, we construct a sequence of cohomology classes for the moduli spaces of smooth 4-manifolds. The corresponding characteristic class can detect the subtle difference between the topological category and the smooth category. And we use them to prove that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology, unlike what happens in all even dimensions not equal to 4. Time permitting, we will also discuss how this characteristic class can detect an infinite-rank summand in the fundamental group of the diffeomorphism group of many 4-manifolds (e.g. all simply-connected elliptic surfaces).

(joint with M. Anel, E. Finster and A. Joyal)

Given a higher topos E and and a left exact localization L we construct a tower of left exact localizations $(P_n)_{n\ge 0}$ of E such that $P_0=L$. We call it the "generalized Goodwillie tower of the pair (E,L)". The construction comes out of a careful study of the factorization systems (modalities) associated to these localizations. The higher stages are described in terms of pushout product powers of its zeroth level. The layers of the tower are stable in the sense that cartesian and cocartesian squares coincide. This is a consequence of a Blakers-Massey-type theorem that comes with each generalized Goodwillie tower. Special cases of the tower are the classical Goodwillie tower, as well as Weiss' orthogonal tower.