Topology Seminars Autumn 2008

29 September
2008 A TOPOLOGIST'S VIEW OF WEIGHTED PROJECTIVE SPACE
Nigel Ray, University of Manchester
400pm in FRANK ADAMS 2Abstract (click to view)Weighted projective spaces are extremely interesting generalisations of standard projective space, and are orbifolds rather than manifolds; they originally arose in algebraic and differential geometry, as well as theoretical physics. They have been little explored by topologists, so I propose to spend some time on their definition, and to set the scene for recent advances concerning their homeomorphism type, and integral and equivariant cohomology. The theory of piecewise polynomials, originally developed by numerical analyists, plays a crucial role.

6 October
2008 SPHERICAL CLASSES IN THE HOMOLOGY OF LOOP SPACES
Hadi Zare, University of Manchester
400pm in FRANK ADAMS 2Abstract (click to view)I will start by mentioning the definition of a spherical class, and saying a few words about their basic properties. Working with the homology of loop spaces, these properties together eliminates a wide range of homology classes from being spherical, although the set of remaining classes is still "too big". I will mention some results on single loop spaces, where it is fairly easy to determine the spherical classes.

13 October
2008 JOINS AND ALEXANDER WHITNEY MAPS
John Jones, University of Warwick
400pm in FRANK ADAMS 2Abstract (click to view)Abstract here

20 October
2008 GALOIS EXTENSIONS OF THE K(n)LOCAL SPHERE
Andrew Baker, University of Glasgow
400pm in FRANK ADAMS 2Abstract (click to view)The notion of a Galois extension of commutative Salgebras (= Einfinty ring spectra) was introduced by John Rognes. While algebraic Galois theory embeds into stable homotopy theory via EilenbergMacLane spectra, there are many more subtle examples than those coming from algebra. For example, KO > KU gives an example of a Galois extension with group C_2. Rognes showed that the sphere spectrum admits no connected Galois extensions with finite Galois group, essentially because there are no unramified Galois extensions of the rationals. On the other hand, the K(n)local sphere S_{K(n)} spectrum admits the LubinTate spectra as Galois extensions with profinite Galois group. I will discuss this and a recent result with Birgit Richter which determines the 'algebraic closure' of S_{K(n)}.

27 October
2008 THE RING OF COMBINATORIAL POLYTOPES AND FORMAL GROUPS
Victor Buchstaber, Universities of Manchester and Moscow
400pm in FRANK ADAMS 2Abstract here

3 November
2008 NO SEMINAR  TTT in LEICESTER
LECTURER
400pm in FRANK ADAMS 2Abstract (click to view)Abstract here

10 November
2008 FROM REAL QUADRICS TO POLYTOPES VIA MANIFOLDS
Taras Panov, Moscow State University
400pm in FRANK ADAMS 2Abstract (click to view)Manifolds obtained as complete intersections of real quadratic hypersurfaces in a complex space have a natural torus action on them, and are known to toric topologists as momentangle manifolds. They correspond naturally to combinatorial simple polytopes, and a direct passage from quadrics to polytopes involves some nice convex geometrical reasoning. The quadratic equations or the polytopes may be very simple, while the corresponding momentangle manifolds usually are quite complicated topologically. Studying their topology proves to be an interesting and challenging problem.

17 November
2008 HYPERSURFACE SPACES (provisional)
John Greenlees, University of Sheffield
400pm in FRANK ADAMS 2Abstract (click to view)Abstract here

24 November
2008 ON THE CASTELNUOVOMUMFORD REGULARITY OF THE COHOMOLOGY OF A GROUP
Peter Symonds, University of Manchester
400pm in FRANK ADAMS 2Abstract (click to view)We will sketch a proof of Benson's conjecture that the regularity of the mod p cohomology of any finite group is 0. First we will discuss what this means. The proof itself uses classical methods: equivariant cohomology and Gysin sequences. There is an obvious connection with my talk in the algebra seminar on 25 November.

1 December
2008 GENERALISATIONS OF QUASITORIC MANIFOLDS
Jerry Hopkinson, University of Manchester
400pm in FRANK ADAMS 2Abstract (click to view)Davis and Januszkiewicz discuss the construction of quasitoric manifolds starting from an $n$dimensional simple polytope with some combinatorial data and a torus of the same dimension, so that the polytope is the orbit space of the $n$torus action on the constructed manifold. Here we extend the constructions to use the group of unit quaternions, and then to a more general class of groups. In each case we discuss which properties (from DJ91) hold for these group manifolds. In particular, we show that the manifolds have a generalised $n$torus action, we discuss the restrictions that are placed on the polytope and combinatorial data, and for which ones we can derive the form of the cohomology ring.

8 December
2008 ABEL FORMAL GROUP LAWS
Francis Clarke, Swansea University
400pm in FRANK ADAMS 2Abstract here

15 December
2008 POINCARE DUALITY ALGEBRAS MOD 2
Larry Smith, Goettingen and Sheffield
400pm in FRANK ADAMS 2Abstract (click to view)This is joint work with Bob Stong. We examine Poincar'e duality algebras over the field of 2 elements which are standardly graded, i.e., generated by their degree one elements. We show that there is a classification of such algebras analogous to the Krull  Schmidt Theorem, i.e., a decomposition of such an algebra into a "connected sum" of "indecomposable" algebras (terms in quotes will be defined in the talk!) and obtain for surface algebras (i.e., PDAs of "formal dimension" 2) the same classificaion that one has for topological surfaces. For higher formal dimension we show that the corresponding Grothendieck group is NOT finitely generated. Finally, we also examine in great detail formal dimension three, obtaining some interesting data for algebras with up to three generators, a quite curious torus bundle over a circle, and encounter some interesting invariant theoretic problems.
Further information
For further information please contact the seminar organiser.