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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 ...


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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 ...


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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:)


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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 $\pm$splitting preserve direct ...


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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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