# What's isotopy of framings?

I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus.At pp.100-101,the authors say "isotopy classes of framings."I don't know how to determine isotopy to framings.Please tell me about this.

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## 1 Answer

Let $$\varphi_0, \varphi_1: V^k \hookrightarrow M^n$$ be two embeddings of the $k$-manifold $V$ into the $n$-manifold $M$, both with trivial normal bundle. Furthermore, let $f_0$ be a framing of the normal bundle $\nu(\varphi_0: V \hookrightarrow M)$, and similarly let $f_1$ be a framing of the normal bundle $\nu(\varphi_1: V \hookrightarrow M)$. Then $(\varphi_0, f_0)$ and $(\varphi_1, f_1)$ are two framed embeddings of $V$ into $M$.

The usual definition of an isotopy between the embeddings $\varphi_0$ and $\varphi_1$ is a homotopy $$F: V \times [0,1] \longrightarrow M$$ between $\varphi_0$ and $\varphi_1$ such that $F(-,t)$ is an embedding for all fixed $t$.

Note that by the Tubular Neighborhood Theorem, a normal framing $f$ of an embedding $$\varphi: V^k \hookrightarrow M^n$$ with trivial normal bundle uniquely determines an embedding $$\tilde{\varphi}: V^k \times D^{n-k} \hookrightarrow M^n$$ up to isotopy, where $D^{n-k}$ is the $(n-k)$-dimensional disk. Then we say two framed embeddings $(\varphi_0,f_0)$ and $(\varphi_1,f_1)$ are isotopic if the induced embeddings $$\tilde{\varphi}_0, \tilde{\varphi}_1: V^k \times D^{n-k} \hookrightarrow M^n$$ are isotopic as embeddings.

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Any chance you know how to find the integer number associated to a framing? – user99680 Feb 2 '14 at 4:41