Schedule for: 23w5017 - Spaces of Manifolds: Algebraic and Geometric Approaches

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Gathering at PDC 2nd Floor BIRS Lounge

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

A century ago, J. W. Alexander showed that the space of homeomorphisms of a disc (which are the identity near the boundary) is contractible, using an explicit radial deformation now known as the "Alexander trick". I will explain recent joint work with Soren Galatius which shows that the same conclusion holds for all compact contractible manifolds of dimension at least 6. In this generality there can be no such explicit deformation, and the question must be approached more obliquely. Along the way, I will explain a new result on spaces of smooth embeddings of one-sided h-cobordisms into other manifolds.

The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope P in R^n, one can cut P into a finite number of smaller polytopes and reassemble these to form Q. Kreck, Neumann and Ossa introduced and studied an analogous notion of cut and paste relation for manifolds called the SK-equivalence ("schneiden und kleben" is German for "cut and paste"). In this talk we will explain the construction that will allow us to speak about the "K-theory of manifolds" spectrum. The zeroth homotopy group of the constructed spectrum recovers the classical groups SK_n. We will show how to relate the spectrum to the algebraic K-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further we will describe the connection of our spectrum with the cobordism category.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk will survey what we know about the rational homotopy theory of the classifying space Baut(M) for fibrations with fiber a simply connected Poincaré duality space M. There are two main points. Firstly, in joint work with Zeman, we have constructed an algebraic model for Baut(M) built out of an arithmetic subgroup of a certain reductive algebraic group together with a dg Lie algebra of algebraic representations of said reductive group. Secondly, I will outline the construction of certain characteristic classes of fibrations associated to cycles in Kontsevich's Lie graph complex and I will discuss some open problems regarding these. This provides a backdrop for the earlier computations of the stable rational cohomology of block diffeomorphism groups of iterated connected sums of products of spheres of Madsen and myself and for the more recent computations of Stoll.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Milnor's prime decomposition theorem says that every connected $3$-manifold $M$ can be written as a connected sum of prime manifolds, uniquely up to diffeomorphism. In joint work with Rachael Boyd and Corey Bregman a homotopy theoretic ``prime decomposition'' of the moduli spaces $B \mathrm{Diff}(M)$, describing them as homotopy colimits of moduli spaces of prime manifolds. I will explain how the moduli spaces of $3$-manifolds assemble into a ``modular $\infty$-operad'' encoding the operation of connected sum. From this perspective our theorem says that this modular $\infty$-operad is freely generated by prime manifolds. I will outline how we use this to prove a conjecture of Kontsevich according to which $B \mathrm{Diff}_{\partial}(M)$ is equivalent to a finite CW-complex whenever $M$ has non-empty boundary, and to compute of the rational cohomology ring of $B \mathrm{Diff}( (S^1 \times S^2) \# (S^1 \times S^2) )$. If time permits, I will also talk about work with Shaul Barkan where prove Hatcher's conjecture on the stable rational homology of $B\mathrm{Diff}((S^1 \times S^2)^{\#g})$ as $g \to \infty$.

A compact manifold M can be forgotten to a simple homotopy type, where the latter notion can be expressed algebraically in terms of Waldhausen A-theory. Replacing A-theory by topological Hochschild homology, we can "invent" the weaker notion of a THH-simple homotopy type. I will explain joint work with Pavel Safronov on how this weaker structure is related to intersection theory on M and how it can be extracted from the (2-truncated) Disk-presehaf of M. As an application we obtain that string topology, namely the loop coproduct, is not homotopy invariant in general. I will also explain partial results, joint with John Klein, in the case where M is closed (and THH is replaced by THR).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Ideas in low-dimensional topology are often powerful also for higher dimensional manifold bundles. For example, Goussarov-Habiro's theory of surgery on 1,3-valent graphs in 3-manifolds can be generalized for higher dimensional manifold bundles. Recently, we further generalized the 3-valent graph surgery of bundles to graphs with valences at most 5 to obtain a chain map from the corresponding part in Kontsevich's graph complex to the singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$. In particular, a nontrivial graph cocycle including the pentagon wheel (with one 5-valent vertex) is mapped to an $(8k-10)$-cycle in $B\mathrm{Diff}_\partial(D^{2k})$. We have a crucial evidence that this cycle is nontrivial in $H_{8k-10}(B\mathrm{Diff}_\partial(D^{2k});Q)$. This is a joint work in progress with Boris Botvinnik.

"A celebrated theorem of Weiss and Williams expresses (in a range of homotopical degrees) the difference between the spaces of diffeomorphisms and block diffeomorphisms of a manifold in terms of its algebraic K-theory. In this talk, I will present an analogous result for embedding spaces. Namely, for M a manifold of dimension at least 5 and P submanifold of M of codimension at least 3, we describe the difference between the spaces of block and ordinary embeddings of P into M as certain infinite loop space involving the relative algebraic K-theory of the pair (M, M−P). The range of degrees in which this description applies is the so-called “concordance embedding stable range” which, by recent developments of Goodwillie–Krannich–Kupers, is far beyond that of the aforementioned theorem of Weiss–Williams. I will also explain how one can use this result to give a full description of the homotopy type (away from 2 and roughly up to the concordance embedding stable range) of the space of long knots of codimension at least 3, that is, embeddings rel boundary of the p-disc into the d-disc for d−p>2 and d>4."

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

One of the significant advances in equivariant stable homotopy theory in recent years occurred in the study of the spectrum of tensor triangulated ideals, a.k.a the Balmer spectrum, of compact G-spectra. This spectrum is now well understood for many groups G, in particular for abelian G. The category of n-excisive functors from Spectra to Spectra is closed symmetric monoidal under Day convolution. As a stable monoidal category, it has many formal similarities to the category of G-spectra. For example, compact objects are dualizable. It therefore seems natural to apply the techniques of tensor-triangulated geometry to the study of the category of functors. In this talk we will describe the Balmer spectrum of the category of n-excisive functors. In the process, we describe the analogue of the Burnside ring for excisive functors, which is π_0 of the endomorphism ring of the identity. The result also requires calculating the Tate blueshift for the symmetric group with respect to the family of non-transitive subgroups. Joint with Tobias Barthel, Drew Heard and Beren Sanders.

For any smooth manifold M of dimension d≥4 we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into M, in every degree that is a multiple of d−3, and show that they are detected in the Taylor tower of Goodwillie and Weiss. In this talk I will explain this construction and relate it to some existing results.

An injective continuous map between Euclidean spaces induces a (derived) map between the little disks operads of the corresponding dimensions. I will present an argument showing that this assignment is a weak equivalence if the codimension is at least three (excluding target dimension 4). This implies that the space of topological embeddings between two manifolds is correctly modelled by the space of derived maps between the corresponding configuration categories, in codimension at least three and up to path component dilemmas. This is in stark contrast to the situation in codimension zero, as we know from Krannich-Kupers. The strategy involves replacing Euclidean spaces by tori, and developing a torus trick for configuration categories. This is joint and long-in-the-making work with Michael Weiss.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe joint work with Igusa, Goodwillie, and Merling on spaces of equivariant smooth h-cobordisms. The main result says that for a finite group G, the space of G-equivariant smooth h-cobordisms on M, stabilized with respect to representation discs, splits apart into non-equivariant h-cobordism spaces. Along the way, we revisit functoriality of the h-cobordism space in the smooth setting, as this is a key ingredient for the splittings. The upshot of this result is an equivariant version of Waldhausen-Jahren-Rognes: the space of stable, equivariant, smooth h-cobordisms can be expressed as a summand of the equivariant algebraic K-theory of spaces.

I will overview joint work with Burklund, Hahn and Zhang on the diffeomorphism classification of high-dimensional, metastably connected, smooth closed manifolds. Our classification is phrased in terms of quadratic algebra in the stable homotopy category.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the computation of K- and L-groups of group rings, two basic infinite groups that one needs to consider are the infinite cyclic group and the infinite dihedral group. The computation of their K- and L-theories are classical results and can be stated, roughly, as follows: There is an excisive approximation from the left which is an equivalence up to an error "Nil" term; furthermore the Nil terms for the infinite cyclic and the the infinite dihedral groups are related, for suitable dualities (the Nil-Nil theorem). We give a generalization of these results to Grothendieck-Witt theory (and other related theories) in the setting of Poincaré categories.

It is a well-known fact that in general, the signature of a total space of a fibre bundle of manifolds is not the product of the signatures of the base and the fibre. However, the signature modulo 4 is multiplicative. One can ask under what assumptions the signature is multiplicative modulo n for n larger than 4. In case the base manifold is stably framed, multiplicativity of the signature predicts that the signature is 0 -- hence, multiplicativity modulo n is the same as asking the signature of the total space to be divisible by n. Chern-Hirzebruch-Serre have shown that the signature of the total space of a fibre bundle of manifolds is given by the signature of the twisted intersection form of the base, the twist is given by the local system of unimodular forms given by the middle dimensional cohomology of the fibre. I will explain how to calculate the integers which appear as the signature of a local system of forms over a stably framed base manifold using Grothendieck--Witt and L-theory and some variants thereof. For manifold bundles (over stably framed base manifolds), this reproduces lower bounds in the divisibility of signatures which have previously been shown by means of index theory, and hence which to the best of our knowledge are a priori special to smooth manifolds bundles, whereas our results apply (in particular) to PD fibrations.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

joint with M.Land, M.Weiss & C.Winges It is a curious fact of life in geometric topology, that the classification of closed manifolds by surgery theory becomes easier as one passes from smooth to piecewise linear and finally to topological manifolds. It was long conjectured that an even cleaner statement should be expected in the somewhat arcane world of homology manifolds of the title, which ought to fill the role of some ""missing manifolds"". This was finally proven by Bryant, Ferry, Mio and Weinberger in the 90's in the form a surgery sequence for homology manifolds, building on an earlier theorem of Ferry and Pedersen that any homology manifold admits a euclidean normal bundle. In the talk I will try to explain this surgery sequence, and further that its existence is incompatible with the result of Ferry and Pedersen. The latter is therefore incorrect and/or the proof the former incomplete.

Embedding calculus is a homotopical technique used to study the space of embeddings between two manifolds. Explicit computations with embedding calculus have proven tractable only after rationalization. Remarkably, the category of spaces admits ''higher rationalizations'' which interpolate between the rational world and the p-local world through ''chromatic homotopy theory''. Goodwillie calculus is known to greatly simplify in this setting, and so it has been speculated that similar simplification occurs in embedding calculus. In pursuit of this goal, we describe an analogy between Goodwillie calculus and embedding calculus which becomes exact after chromatic localization. We then describe how this analogy interacts with Heut's recent characterization of chromatic homotopy theory using spectral Lie algebras.

I will explain an identification of the stable cohomology of the classifying spaces of block diffeomorphisms of connected sums of S^k × S^l (relative to an embedded disk), where 2 < k < l < 2k–1. The result is expressed in terms of a version of Lie graph complex homology, the construction of which I will recall. This also leads to a computation, in a range of degrees, of the stable cohomology of the classifying spaces of diffeomorphisms of these manifolds. In the case l = k+1, this recovers and extends recent results of Ebert–Reinhold.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will describe joint work with Courte, Guillermou, and Kragh on the existence of generating functions for exact Lagrangians. Waldhausen's tube space makes a possibly surprising appearance. I will discuss some open questions that subsequently arise.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.