Thirty Second Meeting of the Transpennine Topology Triangle
Department of Mathematics
University of Manchester
November 19th 2001
Talks will take place in the Maths Tower, Room 9.05; informal discussions
will take place at other times in the Brian Hartley Room, floor 6. We will
meet in the BHR for coffee from about 11.00 am.
Titles and speakers are as follows.
- 11.30 am Frank Neumann (Leicester)
``Etale homotopy and moduli stacks''
- 12.30 pm LUNCH
- 3.00 pm Imma Galvez (Sheffield)
``Elliptic genera and invariants of manifolds with boundary.''
- 4.00 pm TEA (BHR)
- 4.30 pm Thomas Huetteman (Aberdeen)
``Polytopes, Homotopy Colimits, and Algebraic K-Theory''
- "Etale homotopy and moduli stacks"
Using the machinery of etale homotopy theory a la Artin-Mazur we
determine the etale homotopy type of certain moduli stacks over the
rationals parametrizing algebraic curves having fixed finite subgroups
of its automorphism groups, which can be realized in the complex
analytic case as finite subgroups of the mapping class group. After
giving an overview on Deligne-Mumford stacks and how to define its
etale homotopy type, we will show how we can actually determine it in
this concrete algebro-geometric situation via comparison with the
complex analytic case of families of Riemann surfaces with symmetries
and their Teichmueller theory.
- ``Polytopes, Homotopy Colimits, and Algebraic K-Theory''
The aim of this talk is to present a link between combinatorial
topology, homotopy theory, and algebraic K-theory of spaces. My
constructions are motivated by some aspects of toric geometry and
might be considered as a (rather naive) attempt to "do algebraic
geometry over the sphere spectrum".
Given a polytope with integral vertices, there is an associated
(projective) toric variety which comes equipped with a covering by
spectra of monoid rings. Thus the category of quasi-coherent sheaves
is equivalent to a certain diagram category of modules over these
rings. The point is that the covering and the monoids are determined
by the given polytope: the "shape" of the diagrams is given by the
face lattice of the polytope, the monoids make use of the actual
geometry (and not just of the combinatorial structure).
Taking this observation as a starting point, I define an analogous
category of "quasi-coherent homotopy sheaves of topological spaces"
(a certain diagram category of equivariant spaces). The attempt to
analyse its algebraic K-theory leads to the study of certain homotopy
colimits which can be calculated using our knowledge of polytopes.
Reading the analogy between algebra and topology backwards, one can
also define homotopy sheaves in the algebraic setting. I hope that the
improved flexibility of homotopy sheaves (as compared to sheaves in
the usual sense) helps to gain insight into the Quillen K-theory of
Lunch will be taken locally (at the vegetarian "On the Eight Day", for example),
and we expect to visit a local curry house (or similar) early in the
Everyone who wishes to participate is welcome: we shall operate the
usual arrangements for assistance with travel expenses. Please email
firstname.lastname@example.org) if you are interested in attending, so that we can cater for appropriate numbers.
How to get there
If you don't know how to find the Mathematics tower on Oxford Road,
there is more information at
University of Manchester, Department of Mathematics Homepage
The meeting is partially supported by a Scheme 3 grant from the
London Mathematical Society.
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