Degree of an antipodal map Let $f:S^n\to S^n$ a continuous map, $n>0$; we consider the induced homomorphism $f_* : H_n(S^n)\to H_n(S^n)$, and, recalling $H_n(S^n)\simeq\mathbb Z$, define $deg(f)\doteq f_*(1)$.
I'm asked to find the degree of the reflection map $f:(x_0,\dots,x_n)\mapsto (x_0,\dots,x_{n-1},-x_n)$; my main problem is that I should do it only using tools from singular homology theory: I have not dealt with the concept of orientation of a manifold, nor I know anything about reflections on an hyperplane (I've browsed previous questions, and these seems to be the suggested tools to address the quandary).
It looks like the only possible way to proceed is to track the way singular simplices $\sigma:\Delta_n\to S^n$ are influenced by the map $f$, and then try to see how their homology class changes: but it's quite an impervious path, and I don't see how to proceed.
Any hints?
Edit: I'm also having problem dealing with this (apparent) contradiction: consider the maps $f,\pi:S^1\to S^1$ with $f(x_1,x_2)=(-x_1,x_2)$ and $\pi(x_1,x_2) = (x_2,x_1)$; then $f\cdot \pi\cdot f\cdot\pi = -id$, thus apparently $deg(f\cdot\pi))^2=-1$...
 A: If you know a few formal properties of homology, you might argue as follows: 
Realize ${\mathbb S}^n$ as ${\mathbb D}^n / \partial{\mathbb D}^n$ with basepoint $\overline{\partial{\mathbb D}^n}$, you have a pinch map ${\mathbb S}^n \to {\mathbb S}^n\vee{\mathbb S^n}$ collapsing $\{x_n = \frac{1}{2}\}$ to a point, and which you can prove to induce the diagonal ${\mathbb Z}\to{\mathbb Z}\oplus{\mathbb Z}$ on $n$-th singular homology. Moreover, for any two pointed maps $f,g: {\mathbb S}^n\to{\mathbb S}^n$, the map $f\vee g: {\mathbb S}^n\vee{\mathbb S}^n\to{\mathbb S}^n$ induces the map $(\text{deg}(f),\text{deg}(g)): {\mathbb Z}\oplus {\mathbb Z}\to {\mathbb Z}$ on $n$-th homology. Putting both together, you get that the map $f+g:{\mathbb S}^n\to{\mathbb S}^n$ defined as the composition $$f+g:={\mathbb S}^n\to{\mathbb S}^n\vee{\mathbb S}^n\xrightarrow{f\vee g}{\mathbb S}^n$$ has degree $\text{deg}(f)+\text{deg}(g)$. Now, taking $f=\text{id}$ and $g$ the map flipping the $n$-th coordinate, you can check that $f+g$ is nullhomotopic, so ...?
