# M721: Index Theory

## Smooth vector bundles and local coordinates

In this entry we remind the reader of the definition of a smooth vector bundle and give a local coordinates for a smooth section.

Definition [Milnor-Stasheff] A rank n vector bundle is a map $p : E \rightarrow X$ and a vector space structure on $p^{-1}x$ for all $x \in X$ so that ${\forall x \in X, \exists \text { nbhd } U}$ and fiber-preserving homeo ${\phi: p^{-1} U \rightarrow U \times {\bf R}^n}$ so that ${p^{-1}x \rightarrow x \times {\bf R}^n}$ is a v.s. iso. To define a smooth vector bundle one requires that E and X are manifolds and the $\phi$‘s are diffeomorphisms.

Definition [Steenrod (see also Davis-Kirk)] A rank n vector bundle is a map $p : E \rightarrow X$ and a collection of homeos ${\cal B} = \{\phi_i : p^{-1}U_i \rightarrow U_i \times {\bf R}^n \}$ so that

• each ${\phi_i}$ is fiber-preserving over $U_i$.
• ${\{U_i\}}$ is an open cover of ${X}$
• ${\forall i,j, \exists \text{ cont } \theta_{ij} : U_i \cap U_j \rightarrow GL_n({\bf R})}$ so that $\phi_i^{-1}(x,v) = \phi_j^{-1} (x,(\theta_{ij}(x)v)$
• ${\cal B}$ is max’l with respect to the above three properties.
• To define a smooth vector bundle one requires that X is a manifold and the ${\theta_{ij}}$ are smooth.

A smooth section is a smooth map ${s : X \rightarrow E}$ s.t. ${p \circ s = \text{Id}_X}$.

The vector space ${C^\infty(E)}$ of smooth sections is a module over the ring of continuous functions ${C^\infty(X,{\bf R}) \qquad (f,s) \mapsto fs = (x \mapsto f(x)s(x))}$.
e.g. ${C^\infty(X\times {\bf R}^M) = C^\infty(X,{\bf R})^M}$

A smooth section of an ${M}$-plane bundle over an ${n}$-manifold is locally an element of ${C^\infty({\bf R}^n,{\bf R}^M)}$. To make this precise ${\forall x \in X, \exists \text{ nbhd } U}$ and charts ${\phi : p^{-1}U \rightarrow U \times {\bf R}^M}$ and ${h : U \rightarrow {\bf R}^n}$. Then ${C^\infty({\bf R}^n,{\bf R}^M) \cong C^\infty(E|_{U})}$ via ${t \mapsto (u \mapsto \phi^{-1}(u,t(h(u))}$