Euler characteristic with compact support of spaces of Euclidean lattices

Has the Euler characteristic with compact support of $$mathrm{SL}_n(mathbb R)/mathrm{SL}_n(mathbb Z)$$ been computed ? References? Thanks.

I think that the euler characteristic is 0 for the following reasons.

Firstly, the space $$SL_N(mathbb{R})$$ is a bundle over the symmetric space $$SO(N,mathbb{R})backslash SL_N(mathbb{R}) = SP(n,mathbb{R})=X$$, the space of symmetric positive-definite real matrices of determinant 1. For a discussion of this symmetric space, see e.g. Bridson-Haefliger II.10. Then $$SL_N(mathbb{R})/SL_N(mathbb{Z})$$ is a bundle over $$X/SL_N(mathbb{Z})$$ with fiber $$SO(N,mathbb{R})$$. Note that this is an orbifold bundle, but that by passing to a torsion-free subgroup, one can assume that it is a manifold (and since you're interested in euler characteristic, this just multiplies by the index).

Now the space $$X/SL_N(mathbb{Z})$$ admits a bordification by Borel-Serre. Hence $$SL_N(mathbb{R})/SL_N(mathbb{Z})$$ has a bordification by an $$SO(N,mathbb{R})$$-bundle over the Borel-Serre bordification. Hence it is the interior of a manifold with boundary $$M$$. In this case, $$H^*_c(SL_N(mathbb{R})/SL_N(mathbb{Z}))cong H^*(M,partial M)$$. Then by Lefschetz duality, $$chi(H^*_c(M,partial M))=chi(M)$$.

But since $$M$$ is a bundle with fiber $$SO(N,mathbb{R})$$, and $$chi(SO(N,mathbb{R}))=0$$ (any Lie group has a nowhere vanishing vector field), we have $$chi(M)=chi(SO(N,mathbb{R}))times chi(X/SL_N(mathbb{Z})) =0$$, since the euler characteristic of bundles is the product of the euler characteristic of the base and the fiber.

Answered by Ian Agol on November 9, 2021

Related Questions

Sparse perturbation

0  Asked on December 1, 2020 by yiming-xu

What is known about the “unitary group” of a rigged Hilbert space?

2  Asked on November 30, 2020

Odd Steinhaus problem for finite sets

0  Asked on November 30, 2020 by domotorp

Higher-order derivatives of $(e^x + e^{-x})^{-1}$

1  Asked on November 28, 2020 by tobias

Smallness condition for augmented algebras

1  Asked on November 28, 2020 by ttip

number of integer points inside a triangle and its area

1  Asked on November 27, 2020 by johnny-t

Unitary orbits on the Grassmann manifold of 2-planes in complex affine space

1  Asked on November 26, 2020 by norman-goldstein

Should the formula for the inverse of a 2×2 matrix be obvious?

9  Asked on November 21, 2020 by frank-thorne

Hodge structure and rational coefficients

0  Asked on November 19, 2020 by dmitry-vaintrob

Optimal path with multiple costs

2  Asked on November 18, 2020 by lchen

Can a fixed finite-length straightedge and finite-size compass still construct all constructible points in the plane?

2  Asked on November 17, 2020 by joel-david-hamkins

Genus $0$ algebraic curves integral points decidable?

0  Asked on November 14, 2020 by 1

Interlocking (weak) factorization systems

0  Asked on November 9, 2020 by tim-campion

Monte Carlo simulations

3  Asked on November 7, 2020 by alekk

Measurable total order

1  Asked on November 5, 2020 by aryeh-kontorovich

Recover approximate monotonicity of induced norms

1  Asked on November 3, 2020 by ippiki-ookami

Geodesics and potential function

0  Asked on October 29, 2020 by bruno-peixoto

Independent increments for the Brownian motion on a Riemannian manifold

0  Asked on October 26, 2020 by alex-m

Multivariate monotonic function

2  Asked on October 25, 2020 by kurisuto-asutora