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Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively.

Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ are isotopic. We attach spaces $M$ and $N$ via $f$ and $g$ and obtain $M\cup_fN$ and $M\cup_gN$.

I want to prove (or disprove) that $M\cup_fN$ and $M\cup_gN$ are homeomorphic.

First of all, I am confused by the difinition of isotopy. My understanding is that homeomorphisms $f: A \to B$ and $g:A \to B$ are isotopic if there is a map $H: A\times [0,1] \to B$ such that $H(x, t)$ is a homeomorphism for each $t\in [0,1]$ and $H(x, 0)=f(x)$ and $H(x,1)=g(x)$. Is this definition correct in this context?

Any suggestions to the definition of isotpy and construction of a homeomorphism between $M\cup_fN$ and $M\cup_gN$ are appreciated.

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The definition of isotopy is correct. Do you know whether the result is true or not? At least I cannot think of a counterexample.. – M.B. Jun 11 '12 at 18:24
@M.B. I'm not sure if this is true or not. We might meed more conditions on spaces. – Primo Jun 11 '12 at 19:55
So this is your own conjecture? I.e. not some problem from an algebraic topology class? If it actually is true in the topological case it looks difficult to prove. – M.B. Jun 11 '12 at 23:31
up vote 4 down vote accepted

In the differential context (ie, all spaces are smooth manifolds), if $f,g\colon A\rightarrow N$ are smooth maps, an ambient isotopy between $f$ and $g$ is a (smooth) isotopy $F\colon N\times R\rightarrow N$ such that $F(p,0) = p$, for $p\in N$ and $F(f,1)= g$. If $f$ and $g$ are ambient isotopic, then $M\cup_f N$ and $M\cup_g N$ are diffeomorphic.

It is a theorem of Thom, Cerf and Palais that if $A$ is compact and $N$ is closed, then two embeddings are isotopic if and only if they are ambient isotopic (see Theorem 5.2 in chapter II in Kosinski's Differential manifolds).

EDIT: Regarding the topological version of this theorem: Suppose $A = S^1$ and $M = S^3$, and $f,g$ are the trivial knot and the trefoil knot. Then $f$ and $g$ are topologically isotopic, but their complements are not homeomorphic, so this topological isotopy does not extend to an ambient isotopy (In there is an account of topological isotopy, with conditions for a topological isotopy extension theorem to be true and references)

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Do you know if there is any version of this theorem for complex manifolds? – Lor Aug 12 '14 at 16:41

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