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K-State Today

August 25, 2015

Mathematics Colloquium lecture Aug. 25

By Reta McDermott

George Glauberman, University of Chicago, will present "An Application of Group Theory to Topology" as part of the Mathematics Department Colloquium lecture series at 2:30 p.m. Tuesday, Aug. 25, in 122 Cardwell Hall.

The abstract for the lecture is: Let p be a prime. To every finite group is associated a topological space known as the p-completion of its classifying space. The Martino-Priddy conjecture states that for two groups G and H, these spaces are homotopy equivalent if and only if there is a special type of isomorphism between their Sylow p-subgroups (an isomorphism of fusion systems, e.g., elements conjugate in G are mapped to elements conjugate in H).

J. Martino and S. Priddy proved the "only if" part in 1996. B. Oliver proved the converse for odd p in 2004 and p = 2 in 2006. In 2013, A. Chermak proved a strong generalization of the conjecture and Oliver proved an extension of Chermak's result. Each of these four proofs relied partly on assuming the classification of finite simple groups. Recently, J. Lynd and I removed this assumption. I plan to discuss the main ideas of the three recent results.