Topology Seminars 2009/10

18 January
2010 Cupproducts in the cohomology of generalized momentangle complexes.
Tony Bahri
3pm in FRANK ADAMS 2Abstract (click to view)Associated to a simplicial complex K on m vertices and a family of m based CW pairs, is a CW complex called a generalized momentangle complex. After one suspension, this space decomposes into a wedge of spaces determined by the full subcomplexes of K. Paradoxically, this stable splitting can be exploited to give a description of the cohomology ring structure. The results complement cases studied by Hochster, Franz, Panov, BuskakovBuchstaberPanov and Davis Januszkiewicz. A report of joint work with M. Bendersky, F. Cohen and S. Gitler.

1 February
2010 Realisations of the face ring, and homotopy uniqueness
Nigel Ray (University of Manchester)
4pm in FRANK ADAMS 2Abstract (click to view)This talk will focus on joint work with Dietrich Notbohm (VU Amsterdam). In 1991, for any finite simplicial complex K, Davis and Januszkiewicz defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the face ring (or StanleyReisner algebra} of K. In 2002, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to the original examples. Our project is therefore to study the extent to which the homotopy type of a space is determined by having such a cohomology ring. I shall outline our analysis of this problem (i) rationally, and (ii) prime by prime, and then attempt to explain how the outcomes may be reassembled using Sullivan's arithmetic square. The entire problem becomes straightforward after a single suspension, and we shall warmup by discussing this situation first.

8 February
2010 Ring of polytopes and Rota  Hopf algebra.
Victor Buchstaber (University of Manchester)
4pm in FRANK ADAMS 2Abstract (click to view)Nowadays Hopf algebras is one of the wellknown tools in combinatorics.
To study the combinatorics of convex polytopes we develop the approach based on the ring P of combinatorial convex polytopes. We show that the ring of polytopes has a natural Hopf comodule structure over the RotaHopf algebra of posets. As a corollary we build a ring homomorphism l_alpha: P>R[alpha] such that F(l_alpha (P))=f(P)*, where F is the Ehrenborg quasisymmetric function.
The talk is based on the paper: V.~M.~Buchstaber, N.~Yu.~Erokhovets, Ring of polytopes, quasisymmetric functions and Fibonacci numbers., see arXiv February 2010. 
15 February
2010 Transversal homotopy theory
Jon Woolf (University of Liverpool)
4pm in FRANK ADAMS 2Abstract (click to view)Given a smooth stratified space $X$ one can construct new invariants of it by considering maps $S^k\to X$ which are transversal to all strata, up to homotopy through such maps. The resulting invariants $\psi_k(X)$ are dagger monoids, and capture information about the stratification of $X$ as well as the topology of the underlying space. A higher version of this construction associates a rigid monoidal dagger category $\Psi_k(X)$ to each smooth stratified space. The category is braided for $k>1$ and symmetric for $k>2$. As an example, if $X=S2$ stratified by a point and its complement then $\Psi_2(X)$ is equivalent to the category of framed tangles.

22 February
2010 Simplicial structures on braid groups and mapping class groups
Elizabeth Hanbury (Durham University)
4pm in FRANK ADAMS 2Abstract (click to view)This talk will be about braid groups and mapping class groups of surfaces. For a given surface M, the collection of braid groups of M and the collection of mapping class groups of M each carry a simplicial structure. I will explain why this is so, and how these simplicial structures are related to the homotopy groups of spheres.

1 March
2010 A complete $g$vector for convex polytopes
Jonathan Fine (Open University)
4pm in FRANK ADAMS 2Abstract (click to view)Intersection homology defines primitive Betti numbers $g_i(X)$ for every convex polytope $X$ and $0 \leq 2i \leq \dim X$. These numbers are linear functions of the flag vector $f(X)$.
In this talk I extend the combinatorial definition to give $g_k(X)$ that encode the whole of $f(X)$. Computer calculation suggests that for many $k$ we have $g_k(X)\geq 0$. The proof of such results probably requires an extension of intersection homology, that detects singularities.
The talk is based on my arXiv preprint with the same title. 
8 March
2010 On the double loop space on an odd sphere.
Hao Zhao (University of Manchester)
4pm in FRANK ADAMS 2Abstract (click to view) 
15 March
2010 Abelian covers of the real DavisJanuszkiewicz space
Alvise Trevisan (Vrije Universiteit, Amsterdam)
4pm in FRANK ADAMS 2Abstract (click to view)The real DavisJanuszkiewicz space DJ(K) of a simplicial complex K is a particular momentangle construction with a rich combinatorial structure. In this talk I explain how to compute the rational Betti numbers of any regular abelian cover of DJ(K). As a corollary we obtain the rational Betti numbers of the real part of a smooth projective toric variety and, more generally of any real quasitoric manifold.

22 March
2010 The Cayley Plane and the Witten Genus
Carl McTague (Cambridge University)
4pm in FRANK ADAMS 2Abstract (click to view)Elliptic cohomology is at the heart of many recent developments in algebraic topology. (HillHopkinsRavenel for example recently used it to solve the Kervaire invariant problem.) What led to its discovery was Ochanine's observation in the 1980s that there are many more multiplicative genera for spin fiber bundles than for oriented fiber bundles, one for each elliptic curve with a marked point of order 2. Given that multiplicative genera for spin fiber bundles have led to such unexpectedly rich developments, it seems reasonable to investigate multiplicative genera for other types of fiber bundles, in particular O<8> fiber bundles. I will discuss a recently published result of Dessai and a result of my own which in investigating this question place the Witten genus into a geometric framework.

19 April
2010 Homogeneous coordinates on toric varieties
Gregory Sankaran (University of Bath)
4pm in FRANK ADAMS 2Abstract (click to view) 
26 April
2010 Manchester SIAM Student Chapter
no seminar today
4pm in FRANK ADAMS 2Abstract (click to view) 
10 May
2010 Brunnian braids, homotopy and Lie powers
Jie Wu (National University of Singapore)
4pm in FRANK ADAMS 2Abstract (click to view)In this talk, we will discuss the Brunnian braids on general surfaces and their connections with the homotopy groups of spheres. By considering Vassiliev invariants of Brunnian braids, we will also discuss the connections between the Brunnian braids, the symmetric group module Lie(n) and the Lie powers.

20 May
2010 Brunnian braids, homotopy and Lie powers
Jie Wu (National University of Singapore)
12pm in FRANK ADAMS 2Abstract (click to view)In this talk, we will discuss the Brunnian braids on general surfaces and their connections with the homotopy groups of spheres. By considering Vassiliev invariants of Brunnian braids, we will also discuss the connections between the Brunnian braids, the symmetric group module Lie(n) and the Lie powers.
Further information
For further information please contact the seminar organiser.