Algebraic Topology · Problem 2.2.7

Homology — Allen Hatcher

Mathematics — Topology · degree-theory

Problem

For an invertible linear transformation f:Rn→Rnf: \mathbb{R}^n \to \mathbb{R}^n show that the induced map on Hn(Rn,Rn−{0})≈H~n−1(Rn−{0})≈ZH_n(\mathbb{R}^n, \mathbb{R}^n - \{0\}) \approx \widetilde{H}_{n-1}(\mathbb{R}^n - \{0\}) \approx \mathbb{Z} is 1\mathbb{1} or −1-\mathbb{1} according to whether the determinant of ff is positive or negative. [Use Gaussian elimination to show that the matrix of ff can be joined by a path of invertible matrices to a diagonal matrix with ±1\pm 1's on the diagonal.]

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