TITLE: Connection between determinant of $f$ to its induced homomorphism on homology gps
QUESTION [9 upvotes]: It is my homework from Hatcher's book.
It is a problem 7 on section 2.2, stating:
For an invertible linear transformation $f:\mathbb{R}^n \to \mathbb{R}^n$ show that the induced map on $H_n (\mathbb{R}^n, \mathbb{R}^n-{0}) \sim H_{n-1} (\mathbb{R}^n-{0}) \sim \mathbb{Z} $ is identity or -identity according to whether the determinant of $f$ is positive or negative.
Since $f$ is homeomorphism, it seems obvious that induced homomorphism should be identity or -identity. However, I have no idea how it can be connected to the deterimant of a map.
Any comment would be grateful!
Thank you for reading my question!

REPLY [2 votes]: To expand on Mariano's comment:
It is easy to show that $\text{GL}_n(\mathbf{R})$, with the induced topology from $\text{M}_n(\mathbf{R}) \cong \mathbf{R}^{n^2}$, has two path components, consisting of the matrices having positive or negative determinant.
A path from $f$ to $f'$ (as elements of $\text{GL}_n(\mathbf{R})$) yields a homotopy from $f$ to $f'$ (as homeomorphisms of $\mathbf{R}^{n}$). By the homotopy invariance of homology,  we can therefore replace $f$ with any transformation whose determinant has the same sign (thus we are reduced to the cases $f=\text{Id.}$ or $f=\text{a reflection about a plane}$; see countinghaus' answer).