Abstract: Study of the topological and geometric properties of a (Riemannian) manifold by investigating the spectral properties of the geometric elliptic operators, or in general elliptic complexes, is the approach of the spectral geometry. Witten, in his famous paper "Supersymmetry and Morse theory", used the spectral properties of the perturbed de Rham complex, so called Witten complex, to prove the Morse inequalities. In this talk we shall cover his proof. The idea of the proof is to use the approximations of the eigenvalues of the corresponding laplacian. In the next step, we will have an overview on Bismut's proof. Bismut puts Witten's idea in another format. He proves the inequalities by studying the long term behavior of the heat kernel.

Abstract: Homotopy colimit of classifying spaces of abelian subgroups of a finite group $G$ capture information about the commutativity of the group. For the class of extraspecial p-groups of rank at least 2 these colimits are not of the homotopy type of a $K(\pi,1)$ space. The main ingredient in the proof is the calculation of the fundamental group. Another natural question is the complex $K$-theory of these homotopy colimits which can be computed modulo torsion. In contrast to the classifying space $BG$, torsion groups may appear in $K^1$.