Schedule for: 16w2689 - Surgery and Geometry

Note: the Lecture rooms are available after 16:00.

A buffet breakfast is served daily between 7:00am and 9:30am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

This talk is based on joint work with S. Weinberger and G. Yu. Given a homotopy equivalence between two oriented closed manifolds, one can associate a secondary index invariant, called higher rho invariant. In particular, one can verify that the higher rho invariant induces a well-defined map from the structure group (of a given oriented closed manifold) to the analytic structure group (K-theory group of a certain C*-algebra determined by the fundamental group of the manifold). A priori, one does not know whether this higher-rho-invariant map induces a group homomorphism from the structure group to the analytic structure group. In the case where the fundamental group is finite, this was answered positively by D. Crowley and T. Macko. However, the general case remained an open question for quite some time. In my recent joint work with S. Weinberger and G. Yu, we showed that this map is indeed a group homomorphism.

For integers g and n, let V^{2n+1}_{g} denote the g-fold boundary connect-sum of the manifold D^{n+1}\times S^{n}. In this talk I will discuss recent work of mine with Boris Botvinnik where we determine the (co)homology of the classifying spaces BDiff(V^{2n+1}_{g}, D^{2n}) in the direct limit as g approached infinity, in the case that 2n +1 > 7. In particular we identify its homological type with the infinite loopspace QBO(2n+1)< n >. This calculation enables the determination of all characteristic classes for fibre bundles with fibre V^{2n+1}_{g}, for large g. This result can be viewed as an analogue of the Madsen-Weiss theorem for high dimensional manifolds with boundary.

I will discuss some features of the topology of smooth bundles whose fiber is a closed manifold that supports a nonpositively curved Riemannian metric. Specifically, I will show (topological) rigidity results for the associated vertical tangent bundle and a vanishing theorem for the generalized Miller-Morita-Mumford classes. This is joint work with Tom Farrell and Yi Jiang.

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

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!

Factorization homology is a framework for constructing invariants of n-dimensional manifolds, and of E_n-algebras, in the same breath. Its most powerful feature is its manifest functoriality--many deep theorems can be explained or absorbed into its framework, including (1) the fact that Hochschild homology has a circle action, (2) Segal-McDuff type results on configuration spaces, (3) The relationship between Poincare and Koszul dualities, and (4) the construction of fully extended topological field theories from E_n algebras. (In fact, the Baez-Dolan/Hopkins-Lurie cobordism hypothesis has recently been proven by Ayala-Francis using factorization homology). I won't have time to cover nearly all this, but I'll talk about some old joint work with Ayala-Francis showing the basic structural theorems of factorization homology.

The 2-fold product M x M admits a natural involution by interchange. Are there involutions on M x M with fixed point set homeo to M which are inequivalent to interchange? This problem could be reduced to deciding if there exists a 2-stratified space which is stratified homotopy equivalent but not stratified homeo to the orbit space of interchange. Some examples could be produced by using ordinary surgery theory. To deal with the uniqueness part, we need stratified surgery theory. In this talk, we will briefly survey Quinn's blocked surgery and the Stratified surgery theory of BQ type and W type. As an application, we compute the isovariant structure set for cyclic permutation. S^{Z/2-iso}(S^7 x S^7)≅ Z/2 + Z/2 S^{Z/3-iso}(S^8 x S^8 x S^8)≅ Z + Z + Z Information about group action is extracted by analyzing the geometric representatives of the structure set.

The Wall conjecture predicts that every finitely presented Poincare duality group G is the fundamental group of some closed aspherical manifold M (and the Borel conjecture predicts that M is unique up to homeomorphism). Recently Bartels-Lueck-Weinberger solved the Wall conjecture for hyperbolic groups whose boundary is an n-sphere (n>4). In this talk I will discuss an extension of their work to hyperbolic groups whose boundary is a Sierpinski space. This is joint work with Jean Lafont.

We will discuss some examples of the use of surgery theory in studying (i) the existence of finite group actions on “standard” high-dimensional manifolds such as spheres or products of spheres, (ii) the existence of discrete co-compact infinite group actions on $S^n \times R^k$, and (iii) the classification of topological 4-manifolds with a given fundamental group.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A buffet breakfast is served daily between 7:00am and 9:30am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Given a rational Poincare duality algebra A, is there a manifold M whose rational cohomology ring realizes A? The Hirzebruch signature formula provides an obstruction to the existence of such manifold. In the case of A=Q[x]/, a realizing smooth manifold M^n (called a rational projective plane) could only exist in dimensions n=8(2^a+2^b). Sullivan's rational surgery realization theorem provides the necessary and sufficient condition to the existence; the problem boiled down to finding possible Pontryagin numbers satisfying a set of integrality conditions, which can be reduced to a Diophantine equation in our case. Similar techniques can be applied to study the realization of rational Octonionic projective spaces. This is joint work with Jim Fowler.

The Gromov-Lawson surgery theorem builds a connection between surgery theory and positive scalar curvature of manifolds. I will sketch the proof of this theorem. The technique developed by Gromov and Lawson allows one to further analyse how the space of Riemannian metrics of positive scalar curvature behaves under surgery. This will lead to a generalisation of Gromov-Lawson's theorem. At last I will describe how to connect the dimension restrictions coming from the surgery theorems to bordism theory.

We want to discuss a collection of results around the following Question: Given a smooth compact manifold M without boundary, does M admit a Riemannian metric of positive scalar curvature? We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes. We use a refined version, acting on sections of a bundle of modules over a C^*-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class. The talk will present results of the following type: If M has a submanifold N of codimension k whose Dirac operator has non-trivial index, what conditions imply that M does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map? We will present previous results of Zeidler (k=1), Hanke-Pape-S. (k=2), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold. If time permits and we are in the mood for it, we will also or instead discuss to which extend higher signatures of submanifolds are homotopy invariants, in a context similar to the one above.

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.