Moscow-Petersburg seminar on low-dimensional mathematics
September 5, 2003
Vladimir Chernov (Dartmouth)
"Affine linking numbers and their
application to the study of front propagations".
Abstract: The linking number is the classical invariant of
zero homologous objects $N_1$ and $N_2$ in a manifold $M$
such that $dim N_1+dim N_2+1=dim M$. We use the approach
based on Vassiliev-Gusarov invariants to introduce a new
``Affine linking invariant'' that generalizes linking
numbers to the case of arbitrary nonzerohomologous $N_1$
and $N_2$.
The Causality Relation invariant $CR(W_1, W_2)$ of two
fronts is the algebraic number of times one front has
passed through the origin of the other before the other
appeared. We show that affine linking numbers allow one to
reconstruct the value of $CR(W_1, W_2)$ for all manifolds
except of odd-dimensional rational homology spheres without
any knowledge of the propagation law of fronts, and obtain
other results relating affine linking to front propagation.
The construction of the affine linking numbers is based on
our generalization of Chas-Sullivan string homology bracket
of mapping $S^1\to M$ to the case of mapping of garlands
based on any manifold $P$ to $M$.
The talk is based on joint work with Yuli Rudyak, from the University
of Florida Gainesville.