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Abstract: A Lusternik-Schnirelmann cover (L-S cover for
short) of a subspace A of X is a finite cover
A\sseq U_0 \cup U_1 \cup \cdots \cup U_k
where each U_i is
open in X and each inclusion map U_i into X is
homotopic to a constant map. The Lusternik-Schnirelmann category of
A in X is the least k for which an L-S cover
exists, or \infty if there is no such cover; the notation is
\cat_X(A) = k.
If X is a CW complex, then we can look
at the numbers
\cat_X(X_0), \cat_X(X_1) , \ldots , \cat_X(X_n) ,
\cat_X(X_{n+1}), \ldots
which form a weakly increasing
sequence. Interestingly, this sequence is independent of the choice of
CW decomposition of X, and so it is an invariant of the
homotopy type of X.
I will talk about some surprising numerical structure that can be found in this sequence, show how it can be used to prove interesting results, and speculate about further interesting problems.
Some of this was discovered while I was a JWY Instructor at Dartmouth, and some of it was joint work with Nick Scoville and Rob Nendorf. The talk should be comprehensible to anyone familiar with homotopy of maps.
This talk will be accessible to graduate students.
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