Clutching function

A clutching function is a transition map used to glue together locally trivial pieces of a bundle.

Let $M$ be covered by two open sets $U,V\subset M$ such that a bundle is trivial over each piece. For a principal bundle, take $U\times G$ and $V\times G$ and identify points over the overlap $U\cap V$ by

where $g:U\cap V\to G$ is smooth. The map $g$ is the clutching function; it is exactly the transition function (more precisely, a principal bundle transition function ) for the cover $\{U,V\}$.

For a vector bundle with fiber $F$ and structure group $G\subset \mathrm{GL}(F)$, one instead glues $U\times F$ to $V\times F$ by

If one uses more than two open sets, the clutching functions on overlaps must satisfy the cocycle condition on triple intersections; changing trivializations replaces $g$ by an equivalent cocycle .

Sphere case

For $S^n = D^n_+\cup D^n_-$ (two hemispheres), the overlap deformation retracts to $S^{n-1}$. Thus, a clutching function can often be taken as a map

and many classification results reduce to the homotopy class of this map.

Examples

1. Möbius line bundle over $S^1$. The Möbius bundle is obtained by gluing the ends of $[0,1]\times\mathbb R$ via $(0,v)\sim(1,-v)$. Interpreting this as a rank-1 real vector bundle over $S^1$, the clutching data is “$-1$” in $\mathrm{GL}_1(\mathbb R)$, producing a nontrivial bundle.

2. Complex line bundles over $S^2$. Using the hemisphere cover of $S^2$, a clutching function is a map $g:S^1\to U(1)$. The standard family

   $$g_k(e^{i\theta}) = e^{ik\theta}$$

   yields the complex line bundles with first Chern class equal to $k$.

3. The tangent bundle of $S^2$. The nontriviality of $TS^2$ can be exhibited by a clutching description with structure group $SO(2)$, where the overlap map $S^1\to SO(2)$ has degree $2$.