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Please clear my confusion about difference between homotopy, topology and isotopy. The first question is: Is it true two objects are isotopic implies they are topologically equivalent and topologically equivalent implies they are homotopic? Next question: Let C be a circle. Let C1 be C subtracted a set of a point p and C2 be C subtracted a set of a line segment L. Is C1 isotopic,topologically equivalent or homotopic to C2 respectively?

Thank you in advance.

A.O

user50727
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1 Answers1

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If by topologically equivalent you mean homeomorphic then this is stronger than being isotopy equivalent or homotopy equivalent. For example, $\mathbb R^n$ and $\mathbb R^m$ are homotopy equivalent (they are both contractible : homotopy equivalent to a point). But $\mathbb R^n$ is not homeomorphic to $\mathbb R^m$ if $m\not =n$. Being isotopy equivalent is stronger than homotopy equivalent as we put some further restrictions on the isotopy equivalences.

Walid Taamallah
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  • There is no notion of "isotopy equivalence" for topological spaces. One can talk about two topological embeddings (or homeomorphisms) being isotopic. – Moishe Kohan May 14 '14 at 00:09
  • Two embeddings $f$ and $g$ from $X$ into $Y$ are isotopically equivalent, or simply isotopic, if there is a homotopy between $f$ and $g$ in the space of embeddings from $X$ into $Y$.

    Two spaces $X$ and $Y$ are isotopically equivalent or isotopic if there is an embedding of $X$ into $Y$ and one of $Y$ into $X$ such that composition is isotopic to the identity map.

     – Walid Taamallah May 14 '14 at 05:01