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at.algebraic topology – How mighty is homotopy equivalence(preserving all geometric intersection numbers) disagree from a homeomorphism? Answer

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at.algebraic topology – How mighty is homotopy equivalence(preserving all geometric intersection numbers) disagree from a homeomorphism?

This query once more energy breathe foolish love the final put up(deleted). Let me know I’ll delete it.

Let $Sigma$ breathe a floor with out border and $f:Sigmato Sigma$ breathe
a capable homotopy equivalence. Suppose for all closed curves
$alpha,beta:Bbb S^1toSigma$ we have now
$ibig([fcircalpha],[fcircbeta]large)=ibig([alpha],[beta]large)$,
i.e. $f$ preserves all geometric intersection numbers. Is it undoubted that
$f$ is correctly homotopic to a homeomorphism?

Note that if $Sigma$ is a closed floor, then any homotopy equivalence is homotopic to a homeomorphism. So the issue is limpid with out the spare assumption: “geometric intersection number is preserved.”

Also, point to that any homeomorphism of $Sigma$ preserves the geometric intersection quantity.

I’m not positive concerning the time period “proper,” i.e., the issue energy breathe well-posed if one replaces capable homotopy equivalence with strange homotopy equivalence and capable homotopy with strange homotopy. I used the time period capable retaining in intellect the launch floor.

Any ameliorate will breathe appreciated. Thanks in forward.

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