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I saw an argument in another site re the claim that the genus-g surface Sg does not retract to a circle. The argument was that,using/assuming H_1(Sg,Z)=Z^{2g};

and H_1(C,Z)=Z ; Z the integers and H_1(Sg,Z) if there was a retraction r: Sg-->C , then , for i being the inclusion ; r the retraction, then roi would give an isomorphism Z-->Z , which cannot happen with a composition Z-->Z^{2g}.

The counter ** is that we can compose Z-->Z^{2g}-->Z by n-->(0,..,0,n,..)-->n

by injection i_k into k-th component composed with projection pi_k into k-th component.

I think the counter has no valid grounds; but I cannot think of a good argument against

the counter, other than that the composition of non-isomorphism cannot be an isomorphism, since the automorphisms form a group .

Can anyone think of any better counter to the counter ** ?

Let f: Z-->Z be a group isomorphism , then , use that automorphism

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# Genus-g Surface and Retraction to Circle

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