Lusternik-Schnirelmann prize


A. Dranishnikov, M. Katz, and Y. Rudyak proved the inequality

catsys ≤ catLS

for closed oriented 4-dimensional manifolds. Thus, the systolic category is a lower bound for the Lusternik-Schnirelmann category (LS category). The bound is a corollary of the theorem that every manifold with LS category 2 either is a surface, or has free fundamental group. It would be interesting to study the discrepancy between the two categories in dimension 4.


In more detail, the classical Lusternik-Schnirelmann category catLS(M) of a closed 4-manifold M is an integer between 1 and 4. The systolic category catsys(M) of M was defined in 2006 by M. Katz and Y. Rudyak. The equality of the two categories has been verified both for 2-dimensional complexes and for 3-manifolds, whether orientable or not. Simply connected 4-manifolds satisfy the equality   catsys = catLS, as well.

In the hallowed tradition of P. Erdös, we are hereby offering a prize of $50 for an example showing a strict inequality between the two categories for an orientable closed 4-manifold. For European contestants only, the prize is upped to €50. Weigh also other prizes.

See the recent monograph for further discussion.




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