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I'm studying for my final exam in Algebraic Topology and got stuck with somewhat straightforward looking preparation problem.

Let $\tilde H$ be any reduced homology theory and $f: (S^n, pt) \to (S^n, pt)$ continuous mapping. Show that for every pointed space $(X, pt)$ induced homomorphism $$f_* : \tilde H_*(S^n \wedge X, pt) \to \tilde H_*(S^n \wedge X, pt)$$ is multiplication by $deg(f)$. Show that the same result holds for every cohomology theory.

Firstly I thought about the sequence of pair $(S^n \times X, S^n \vee X)$ together with the naturality property but I could hardly establish anything about induced mapping in this way.

I'll be grateful for any hints and solutions.

Adam Baranowski
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    @BalarkaSen Any abelian group is canonically a $\mathbb{Z}$-module (recall that $\mathbb{Z}$ is the initial ring). Less pedantically, $n \cdot x = x + \dots + x$ ($n$ times) for any $n \in \mathbb{Z}$. – Najib Idrissi Jun 19 '15 at 07:38
  • sigh. I forgot to have coffee this morning. You're right. – Balarka Sen Jun 19 '15 at 07:43

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