The Homology Groups Of G_n,m

Homology groups is a kind of simpler nonnumerical topological invariants in topological space, how to compute the homology or cohomology groups effectively is a very meaningful issue in the topology. Witten proposaled a brand-newly way to compute the homology groups of a manifold from the perspective of physics.In this paper, we computed the homology groups of real Grassmann manifold Gn,2by Witten’s method. At first, we constructed a Morse function f of Gn,2and got the critical points of f with the corresponding Morse indexes. Then we computed the negative gradient vector field-▽f according to the suitable Riemannian metric of Gn,2and got the invariant manifolds of the dynamical system on Gn,2. Further, we found all connecting orbits of two critical points. At last, we defined the Witten’s boundry operator and computed the homology groups of Gn,2by Witten’s method, it also provided a nontrivial example of this method.