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This is an exercise from Algebraic Topology book of Hatcher:

Exercise 2.1.20, pg. 132: Show that $\tilde H_n(X) ≈ \tilde H_{n+1}(SX)$ for all $n$, where $SX$ is the suspension of $X$.

This question is extensively discussed on Natural isomorphism $\tilde H_i(X) \xrightarrow{\cong} \tilde H_{i+1}(\Sigma X)$ where $\Sigma X$ is the suspension of $X$. and Reduced Homology on unreduced suspension.

I offer a somehow alternative proof, which can be geometrically checked by the picture of the suspension (of the circle):

Note that the $SX$ can be realized as the union of two cones $CX_N$ and $CX_S$ with their bases identified.

For the pair $(SX, CX_N)$, we have a long exact sequence: $$ \ldots \to H_i(CX_N) \to H_i(SX) \xrightarrow{f} H_i(SX,CX_N) \to H_{i-1}(CX_N) \to \ldots$$ Then $f$ is clearly an isomorphism since $CX_N$ is contractible.

By Excision Theorem, the inclusions $(SX-N, CX_N- N) \hookrightarrow (SX, CX_N)$ induce isomorphisms $$H_i(SX-N,CX_N- N) \cong H_i(SX, CX_N)$$

Also for the pair $(SX-N,CX_N- N)$, consider the long exact sequence $$\ldots \to H_i(SX-N) \to H_i(SX-N, CX_N -N) \xrightarrow{g} H_i(CX_N -N) \to H_{i-1}(SX-N) \to \ldots$$

Since $SX -N$ is contractible, $g$ is also an isomorphism.

Now, we are done by gathering isomorphisms $f$ and $g$ with the fact $CX_N -N \simeq X$.

Is there any gap in the proof?

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1 Answer 1

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Your proof is essentially correct. However, you have to use three facts:

(1) There exists a long exact sequence for the reduced homology groups, and this has to be used in your proof.

(2) $\tilde{H}_n = H_n$ for $n > 0$.

(3) $\tilde{H}_0(\ast) = 0$.

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