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I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I can't understand the 2nd-paragraph of p.101, where they explain framings on the attaching sphere. In particular I cannot understand the sentence "By composing $\varphi$ with a self-diffeomorphism of the second factor of $D^{k}\times D^{n-k}$, we can arrange for [an element of $GL(n-k)$] to be the identity at a preassigned basepoint in $S^{k-1}$." I can't understand why there is such a diffeomorphism. Please explain to me how to solve my problem.

In addition to this,I don't exactly know the definition of "framing." In my understanding this, it is an identification of a normal bundle with the trivial bundle.Is this correct?

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    You should say what $D^k$ means. Is it the $k$-dimensional disk? As per "framing", it usually means a choice of global sections that trivialize the bundle under consideration (probably the tangent bundle).2012-04-15
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    I think in this case, a "framing" is probably an embedding of a tubular neighborhood of the thing being surgured (in this case, probably a $k$-disk or a $(k-1)$-sphere).2012-04-15
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    I editted the question a bit. 失礼しました。2012-04-15

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