Transition fuctions only depend on homotopy

Question

I have already posted this question on the physics forum but it didn’t get any answers, hopefully here it will. In the nLab article about Instantons in the section about the clutching construction, it is stated that one can construct $SU(2)$-bundles over $S^4$ via Chech cohomology choosing appropriate transition functions $g_{ij}:U_i\cap U_j \rightarrow SU(2)$ for an appropriate covering $\{U_i\}$ of $S^4$. In the article it is chosen the covering given by two open sets $U_+:=S^4\setminus${point at infinity} and $U_-:= ${tiny neighbourhood of the point at infinity}. Later in the article it is stated than "one finds that instantons are classified by homotopy classes of maps $g:S^3\rightarrow SU(2)$" but I can't understand why. Somehow only the information about the homotopy of the intersection of the two open sets is enough. Can anybody help me?

Answer 1

I'll begin with some general comments on the clutching construction. One can create $G$-bundles over a nice topological space $X$ through the "clutching construction". This is done by specifying the data of some $G$-bundles over an open cover $\{U_i\}_{i\in I}$, that is, for each $i$ in $I$ we have a $G$-bundle $P_i\to U_i$. In order to create a global object on $X$, we need a way to patch (or clutch) together the data we just provided. This is the data of isomorphisms $\psi_{ij}: P_i\vert_{U_i\cap U_j}\to P_{j}\vert_{U_i\cap U_j}$ which respect restriction, i.e. $\psi_{jk}\circ \psi_{ij}=\psi_{ik}$ (with the appropriate changes in domain). We then construct the total space of a $G$ bundle, via this data by $P:= (\bigsqcup_{i} P_i)/(x\sim\psi_{ij}(x))$.

If $X$ is locally contractible, i.e. it admits an open cover by contractible neighborhoods, we can take such a cover $\{U_i\}_{i\in I}$. Now, the homotopy theory of $G$-bundles tells us that any $G$-bundle over a contractible set is necessarily trivial, so we can take each $P_i=U_i\times G$. This then means that each of the transition functions gives a map $U_{i}\cap U_j\to G$ (automorphisms of a trivial $G$-bundle are given by $(u, g)\mapsto (u, \phi(u)g)$). Any homotopy of clutching functions $\{\psi_{ij}^t\}_{i,j\in I^2}$, meaning a smoothly varying collection of functions $\psi_{ij}^t: U_{i}\cap U_{j}\to G$ which satisfy $\psi_{jk}^t \psi_{ij}^t=\psi_{ik}^t$ for all $t$, determine a vector bundle $E_t$ over $[0,1]\times X$ since the $\{[0,1]\times U_i\}$ forms an open cover for $[0,1]\times X$. Since $[0,1]$ is contractible, it follows that the restrictions $E_t\vert_{\{0\}\times X}$ and $E_t\vert_{\{1\}\times X}$ are isomorphic as vector bundles over $X$. By construction, $E_{t}\vert_{\{\tau\}\times X}$ is the bundle that we get by taking the clutching functions $\{\psi_{ij}^\tau\}$ for fixed $\tau$.

Now we turn our attention to the sphere. Let $P\to S^n$ be a principal $G$ bundle. We can cover $S^n$ by the open sets $U^{\pm}=S^n\backslash{(\pm 1, 0, 0,\cdots, 0)}$, which are diffeomorphic to $\mathbb{R}^n$ via stereographic projection and are hence contractible. Since this cover has no triple intersections, we do not have to verify any cocycle condition and we may conclude that any principal $G$-bundle over $S^n$ is gotten by specifying clutching functions $\psi_{+-}: U^+\cap U^-\to G$ and any homotopic map yields an isomorphic $G$-bundle. Because the intersection $U^+\cap U^-$ deformation retracts to the equatorial $S^{n-1}$ of $S^n$, all homotopy classes of maps $U^+\cap U^-\to G$ are represented by a map $S^{n-1}\to G$. The retraction $p:U^+\cap U^-\to S^{n-1}$ is given by $(x_0, x_1,\cdots, x_n)\mapsto \frac{1}{\sum x_i^2}(x_1, \cdots, x_n)$ so we may produce the proper clutching function by taking $\tilde{f}=f\circ p: U^+\cap U^-\to G$. $$\tilde{f}(x_0, \cdots, x_n)=f\left(\frac{x_1}{\sum_{i} x_i^2}, \frac{x_2}{\sum_{i} x_i^2}, \cdots, \frac{x_n}{\sum_{i} x_i^2}\right)$$