分野別セミナー

Among the most basic invariants of a topological space are its Betti numbers, defined as the dimensions of its homology groups, e.g. (1,2g,1) for a surface of genus g and (1,0,...,0,1) for an n-dimensional sphere. Surprisingly, the very simple question "What are the Betti numbers of a manifold?", i.e., the question of what Betti numbers can occur, is not at all easy and is not solved completely even for the simplest non-trivial case: for which integers nis there an oriented n-manifold with total Betti number 3, i.e. with i-th Betti number equal to 1 for i=n/2 and 0 for all other 0