Tuesday Seminar on Topology

Seminar information archive ~04/16Next seminarFuture seminars 04/17~

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya

2007/12/11

16:30-18:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Xavier G\'omez-Mont (CIMAT, Mexico) 16:30-17:30
A Singular Version of The Poincar\\'e-Hopf Theorem
[ Abstract ]
The Poincar\\'e-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.

A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincar\\'e-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincar\\'e-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.
Miguel A. Xicotencatl (CINVESTAV, Mexico) 17:40-18:40
Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology
[ Abstract ]
(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)

At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.