Mathematics Colloquia and Seminars

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Second talk: Weak homotopy types of finite posets Wednesday July 31st, 11-12pm The order complex K(P) of a poset P is the simplicial complex of chains in P. A result of M. McCord shows that the finite space P and the polyhedron K(P) have the same homology and homotopy groups. Conversely, the weak homotopy type of any compact polyhedron can be realized by a finite poset. Although homotopy types of finite spaces are easy to describe with Stong’s ideas, weak homotopy types turn out to be far more difficult to study (algorithmically undecidable). We will show the relationship between this problem and Whitehead’s simple homotopy theory. Third talk: The poset of p-subgroups of a group Wednesday July 31st, 12-1pm. If G is a finite group and p is a prime which divides the order of G, we define the poset S_p(G) of non-trivial p-subgroups of G. If G has a non-trivial normal p-subgroup, the order complex of S_p(G) is contractible. The converse is an open problem conjectured by Quillen in 1978. We will see that this question is directly related to the distinction between two classes of finite posets: those which are contractible and those with trivial homotopy groups.