# The Homology Groups of $G_{n,m}(\mathbb{R})$

## TL;DR

This paper computes the homology groups of real Grassmann manifolds using the Witten complex, providing explicit algebraic topological invariants for these spaces.

## Contribution

It offers a novel computation of homology groups for real Grassmann manifolds via the Witten complex, advancing understanding of their topological structure.

## Key findings

- Homology groups of $G_{n,m}(\mathbb{R})$ explicitly calculated
- Application of Witten complex to real Grassmannians demonstrated
- New algebraic invariants for real Grassmann manifolds obtained

## Abstract

In this article, we computed the homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ by Witten complex.