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I was trying to prove that the degree of the suspension $\Sigma f$ of a map $f: S^n \longrightarrow S^n$ is the same as the degree of $f$. I used Mayer-Vietoris to obtain the following diagram:

$$ \require{AMScd} \begin{CD} H_{n+1}(\Sigma S^n) @>{(\Sigma f)_*}>> H_{n+1}(\Sigma S^n) \\ @V\partial VV @VV{\partial}V \\ H_n(S^n) @>>{f_*}> H_n(S^n) \end{CD} $$

Where $\partial$ is the connecting homomorphism in the Mayer-Vietoris sequence. Intuitively this diagram should commute and the claim follows directly from this. However, I am struggling to conclude that the diagram commutes. I read somewhere that the "construction is functorial, so the diagram commutes", but what does this mean exactly?

Thank you:)

mNugget
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1 Answers1

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The statement is that the suspension isomorphism is natural (in the sense of category theory), see e.g. this mathstacksexchange post here. You can of course also find this statement in most books on algebraic topology, such as Hatcher's book.

RobertMuller
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