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You can normalize every automophism in this way: If we restrict the isomorphism $h_a \colon E_a \to E \times D_a$ to the part of $E_a$ over $X \times \{1\}$, we obtain an isomorphism $E \to E \times \{1\}$ sending a vector $v \in E$ to a pair $(g(v),1)$ in $E \times \{1\}$, where $g$ is an isomorphism $E \to E$. Now compose $h_a$ with the isomorphism ...

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I'm no expert on the matter and this was intended as a comment but it ended up to be too long. I think what is meant in the last sentence of the first paragraph you cited is that $h_0h_{\infty}^{-1}$ induces a clutching function $f$ for $E'$ by restricting to $E\times S^1$. The last sentence in the second paragraph seems a bit strange to me... As far as I ...

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I took sometimes to sketch the diagrams needed. I will not write down a complete proof, I just sketch the main passages and then is up to you to fill the details (easy, don't worry). So first of all, note that fact 2 let us define a isomorphism between $\tilde{K}(X/A)$ and $\tilde{K}(X)$. In fact it preserve the direct sums (note that the pullback of the ...

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Below by $B^n A$ I mean the Eilenberg-MacLane space $K(A, n)$. The Chern character isomorphism tells you how to compute rational K-theory; in this case it gives $$K^0(\Sigma) \otimes \mathbb{Q} \cong H^0(\Sigma, \mathbb{Q}) \times H^2(\Sigma, \mathbb{Q}) \cong \mathbb{Q}^2$$ $$K^1(\Sigma) \otimes \mathbb{Q} \cong H^1(\Sigma, \mathbb{Q}) \cong ... 1 Yes, that's correct; we can only conclude that complex vector bundles over odd spheres are stably trivial, but not that they're trivial. Using the version of Bott periodicity which states that the homotopy groups \pi_i(U) of the stable unitary group are 2-periodic, we see this as follows. Recall that for any topological group G and any n \ge 2 we ... 1 I'll give you some hints about the prop. You have to prove two facts fact 1 The clutch construction preserve direct sum fact 2 The \pm splitting is unique and preserve the direct sum Using these facts you can easily prove your statement here you can find one of my old answer with some calculation done:) 1 Well, I've put a bounty, but it seems that I've found a solution by myself. So here it is, for future reference :) You'll need these facts, which I don't prove here. Fact 1 The clutch construction over X \times S^2 preserve direct sum:$$[E_1 \oplus E_2 , f_1 \oplus f_2 ] \approx [E_1, f_1] \oplus [E_2,f_2]$$Fact 2 the \pmsplitting preserve direct ... 2 The equation you write \tilde{K}(X)\cong K(X)/\mathbb{Z} is confusing, it should be$$K(X)=\mathbb{Z}\oplus\tilde{K}(X)$$its a small thing but I think its the source of your difficulty. \tilde{K}(X) is the kernel of the degree map$$\deg:K(X)\rightarrow \mathbb{Z} now $H$ has dimension $1$ so its not in the kernel, but $H-1$ has dimension zero so ...

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1) it is $H=E_z$ and so $H\otimes H=E_{z^2}$. 2) I think it is the usual Matrix product. The pointwise refers to points on $S^{k-1}$. For every point $x\in S^{k-1}$ you obtain two matrices by applying f and g. The product clutching function evaluated at any point should be the product of these two. 3) works fine as well if you take the usual matrix product ...

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