M721: Index Theory

Ruminations of a Graduate Class

Smooth vector bundles and local coordinates

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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))}

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    Written by jfdavis

    August 27, 2012 at 1:30 am

    Posted in Uncategorized

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