Flat cycles in the homology of \Gamma \SL(m,R)/SO(m)

TL;DR

This paper demonstrates that flat (m-1)-dimensional tori produce nontrivial rational homology cycles in congruence covers of the space SL(m,Z) ackslash SL(m,R)/SO(m), with the dimension of this subspace increasing in higher covers.

Contribution

It establishes the existence and growth of flat torus cycles in the homology of congruence covers of certain locally symmetric spaces, revealing new topological features.

Findings

Flat (m-1)-tori generate nontrivial homology cycles.

The dimension of the homology subspace spanned by these tori increases in higher congruence covers.

The results connect geometric structures with algebraic topology in arithmetic locally symmetric spaces.

Abstract

In this paper we show that flat (m-1)-dimensional tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL(m,Z) \SL(m,R)/SO(m). We also show that the dimension of the subspace of H_{m-1}(\Gamma \SL(m,R)/SO(m);Q) spanned by flat (m-1)-tori grows as one goes up in congruence covers.