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Let $C$ be a compact manifold and $f,g$ two open, continuous embeddings $f,g: C\times [0,1)^r \to M$ where $M$ is a topological manifold with boundary. If $f$ and $g$ are equal on $C\times \{0\}$, can I say there is an isotopy $h$ from $f$ to $g$ on a neighborhood $U$ of $C\times \{0\}$ in $C\times [0,1)^r$?

If $r=1$, this is just the isotopy of collars. I've thought about using Edward's & Kirby's extension of isotopy but thought there might a nicer way of putting it. Or is there a well-known counter-example?

Tom Leness
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A map $S^1 \times [0,1)^2 \to \mathbb R^3$ has a $\mathbb Z$-valued isotopy invariant, namely the linking number of the image in $\mathbb R^3$ of the two circles $S^1 \times \{(0,1/2)\}$ and $S^1 \times \{(1/2,0)\}$. For a fixed map on $S^1 \times \{(0,0)\}$, any integer can be achieved, so there are infinitely many isotopy classes.

Lee Mosher
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