In Lecture 3, we learned about a combinatorial method for representing spaces. In Lecture 4, we studied groups and equivalence relations implied by their normal subgroups. In this lecture, we look at a combinatorial and computable functor called homology that gives us a finite description of the topology of a space. Homology groups may be regarded as an algebraization of the first layer of geom...

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in [PZ2]. Generalized Harish-Chandra modules are (g, k)-modules of finite type where g is a semisimple Lie algebra and k ⊂ g is a reductive in g subalgebra. A first result of the present paper is that a fundamental series module is a g-module of finite length. We then define the notions of strongly a...

We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations C2, using ordinary graded on fundamental groupoid, with coefficients in Burnside ring Mackey functor. This extension RO(C2)-graded theory allows for definition Euler classes, which are used as generators spaces. As an application, we give version Bézout's theorem.

In Lecture 3, we learned about a combinatorial method for representing spaces. In Lecture 4, we studied groups and equivalence relations implied by their normal subgroups. In this lecture, we look at a combinatorial and computable functor called homology that gives us a finite description of the topology of a space. Homology groups may be regarded as an algebraization of the first layer of geom...

In this paper we introduce and study a cohomology theory {Hn(−,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)}n≥0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−, n) is right adjoint to the functor ℘n, w...