Question regarding the paper by Atiyah, Bott and Shapiro: alternative description of K-theory

Question

In Atiyah, Bott, and Shapiro - Clifford modules (journal, MSN), the authors discuss the alternative description of K-theory in terms of sequences of vector bundles. I would like to understand the details of this alternative description and I came across the following question while reading this paper:

In the proof of Lemma 7.3 authors claim that (using the notation from the paper) if we take two monomorphic extensions $\sigma_{n+1}',\sigma_{n+1}''$ of $\sigma_{n+1}$ then $E_n' \cong E_n''$ where $E_n'$ and $E_n''$ are cokernels of $\sigma_{n+1}'$ and $\sigma_{n+1}''$. It is true that these two extensions are homotopic rel $Y$ (this follows from the previous lemma). How does it follow that $E_n' \cong E_n''$?

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Answer 1

The image of the homotopy defines a subbundle of the bundle $E_n\times I$ over $X\times I$. The quotient bundle restricts to $E'_n$ over $X\times 0$ and $E''_n$ over $X\times 1$ and, in general, if you have a vector bundle over $X\times I$ then the restrictions to the two ends are isomorphic.