Arnold' seminar, Moscow University
Tuesday, September 16, 2003

Vladimir Chernov (Dartmouth)
"Affine linking and winding numbers and their 
applications to the study of causality and of front 
propagations" 

(based on joint work with Yuli Rudyak, 
from the University of Florida Gainesville). 

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. 

We use similar ideas to obtain an affine generalization of 
the winding number invariant $win$ to the case of arbitrary 
manifolds and arbitrary hyper-surfaces in them. The affine 
winding number $win$ appears to be very useful to calculate 
the algebraic number of times a wave fronts has passed 
through a point between two time moments. 

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 $M$. We also generalize many 
of other Chas-Sullivan algebraic operations on string 
homology to the case of bordism groups of mappings of 
garlands based on arbitrary manifolds. 

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