Proving bijection between homotopy classes of clutching functions and vector bundles

Question

I am reading Hatcher's book on Vector Bundles and K-Theory(freely available online). I am reading about clutching functions in the first chapter and there is a step in the proof of Proposition 1.11 (p.23) that I don't understand. In order to be concrete I introduce a capture:

I summarize the steps of the proof (as I understand them):

1. In order to show there is a bijection we will construct and inverse $\psi$ to $\phi$. Remember that $p\colon E\to \mathbb{S}^k$ is our vector bundle.
2. Given two trivializations (given by the two hemispheres) $h_+\colon E_+ \to D^k_+\times \mathbb{C}^n$ and $h_-\colon E_- \to D^k_-\times \mathbb{C}^n$, we define $$\psi(E)\in [\mathbb{S}^{k-1},GL_n(\mathbb{C})]$$ to be the homotopy class of the map $\mathbb{S}^{k-1}\to GL_n(\mathbb{C})$ defined by $${h_+}\circ {h_-}_{|\partial D^k_- \times \mathbb{C}}={h_+}\circ {h_-}_{|\mathbb{S}^{k-1} \times \mathbb{C}}$$
3. We have to prove that the map $\psi$ is well-defined and that $\psi$ and $\phi$ are inverses to each other.

My problems arise when trying to prove that $\psi$ is well-defined. So let's begin and I will point out the step I don't understand:

1. Given other trivializations over the hemispheres $h'_+\colon E_+ \to D^k_+\times \mathbb{C}^n$ and $h'_-\colon E_- \to D^k_-\times \mathbb{C}^n$, we define $$\psi(E)\in [\mathbb{S}^{k-1},GL_n(\mathbb{C})]$$ to be the homotopy class of the map $$\mathbb{S}^{k-1}\to GL_n(\mathbb{C})$$ defined by $${h'_+}\circ {h'_-}_{|\mathbb{S}^{k-1} \times \mathbb{C}}$$ We have to check that the two maps: $${h_+}\circ {h_-}_{|\mathbb{S}^{k-1} \times \mathbb{C}}$$ $${h'_+}\circ {h'_-}_{|\mathbb{S}^{k-1} \times \mathbb{C}}$$

are homotopic. And here is where I lost Hatcher's argument. I see that both $h_+\colon E_+ \to D^k_+\times \mathbb{C}^n$ and $h_-\colon E_- \to D^k_-\times \mathbb{C}^n$ are unique up to homotopy. But in order to prove that $${h_+}\circ {h_-}_{|\mathbb{S}^{k-1} \times \mathbb{C}}$$ is unique up to homotopy I think I need to show that the restrictions of the trivializations:

$${h_+}_{|\partial D^k_+ \times \mathbb{C}}$$ $${h_-}_{|\partial D^k_- \times \mathbb{C}}$$

are unique up to homotopy and here the argument provided does not work since the domains of the restriction are not contractible.

I am sure that there is something here I don't understand properly. So any clarification, explanation or help would be appreciated. Thanks!

Answer 1

It is explained here. The key step is to use the notion of equivalent cocycles.