Lecture 6 - Thursday Week 3
Vector Bundles
[edit]Vector Bundles $\pi: E \rightarrow M$ and their sections $\Delta: M \rightarrow E$. We have the notation $\Gamma(E) = \{\Delta | \Delta \text{section of }E\}$
If $\dim(M) > 0$ then $\Gamma(E)$ is infinite-dimensional $\mathbb{R}$ vector space. Also $\Gamma(E)$ is a $C^\infty(M)$-module.
Prop (Ex): If $M$ paracompact, then for all vector bundles $\pi:E \rightarrow M$, there exists $\tilde{E} \rightarrow M$ (finite rank vector bundle) sch that $E \oplus \tilde{E} \rightarrow M$ is trivial. $E \oplus \tilde{E} \cong M \times k^N$ for some $N \in \mathbb{N}$.
def: an $R$-module $\epsilon$ is projective if there exists $\tilde{\epsilon}$ such that $\epsilon \oplus \tilde{\epsilon } \cong R^N$.
Theorem (Serre-Swan): vector bundles over $M$ $\leftrightarrow$ projective modules $M$ over $R = C^\infty(M)$. \textit{Note: we won't really use this theorem.}
prop: If $E$, $F$ are vector vundles, then can take:
- direct sum $E \oplus F$
- duals $E^*$
- tensor product $E \otimes F$
- exterior product $\wedge^k E$
- symmetric products sym$^k E$
def: A subbundle $E \subset F$ is a vertical morphism $E \rightarrow F \rightarrow M$ that is injective everywhere. (note there shold be a commutative diagram here including the map $E \rightarrow M$)
e.g. if $f: M \rightarrow N$ embedding (or immersion) then $TM \hookrightarrow f^* TN$ is a subbundle over $M$.
prop: If $E \hookrightarrow F$ is a subbundle, then can take quotient bundle $F/E$.
e.g. if $f: M \rightarrow N$ immersion, then $\text{Norm}_{M/N} = f^*(TN)/TM$
(Lie) group actions
[edit]def: A (left) action of Lie Group $G$ on a smooth manifold $M$ is given by a map (smooth morphism) $\Psi G\times M \rightarrow M$ given by $(g, m) \mapsto g \cdot m$ such that
- $\Psi(1, m) = m$ for all $m \in M$
- $\Psi(g, \Psi(h,m)) = \Psi(gh, m)$ for all $g,h \in G$ and $m \in M$.
For all $g \in G$ we have diffeomorphism $\Psi_g: M \rightarrow M$ taking $m \mapsto \Psi(g, m)$.
This gives us a group morphism $G \rightarrow \text{Diffeo}(M)$.
Similarly, we define right action.
def: $m \in M$, and $G$ acting on $M$, then $\text{stab}_m = \{m \in M | g \cdot m = m\}.$
def: a group action is:
- effective if $\bigcap_{m \in M} \text{stab}_m = \{\text{id}\}$. i.e. $G \rightarrow \text{Differ}(M)$ is injective.
- free if $\text{stab}_m = \{\text{id}\}$
- transitive if it has a single orbit
- proper if $\Psi_{\text{ext}} : G \times M \rightarrow M \times M$ given by $(g, m) \mapsto (\Psi(g, m), m)$ is proper.
prop: orbits of lie group actions are embedded closed submanfolds.
prop: if $G$ acting on $M$ is free and proper, then there exists a canonical structure of a smooth manifold on the orbit space. $M / G$ such that $M \rightarrow M / G$ is a smooth morphism.
Fibre Bundles
[edit]def: Let $F, M$ be manifolds. Then a fibre bundle of type $F$ (or an $F$ fibre bundle) is a manifold $E$ with a map $E \rightarrow M$ such that there exists an open cover of $M$ and morphisms from each preimage $\chi_\alpha : \pi^{-1}(u_\alpha) \rightarrow U_\alpha \times F$, the triangle commutes ($\pi = \text{pr}_1 \circ \chi_\alpha$).
terminoligy:
- total space
- base
- proj
- sections
remark: sections may not exist, but local sections always exist
From a local trivialisation, we can construct a fibre bundle by glueing local trivialisations using transition functions. $\rho_{\beta\alpha}: U_\alpha \cap U_\beta \rightarrow \text{Diffeo}(F)$ with
- $\rho_{\beta\alpha} = \rho_{\alpha\beta}^{-1}$
- cocycle condition.
A vector bundle is a fibre bundle where each fibre is a vector space. These are the ones we will be most interested in, where $F$ has extra structure and the trivialisations preserve this, and so the transition functions preserve this extra structure. e.g. for vector spaces, where the transition functions $\rho_{\beta\alpha} \subset \text{GL}(V) \subset \text{Diffeo}(V)$.
Principal Bundles
[edit]def: Let $G$ be a lie group. a $G-$principal bundle $P$ (notation $G \rightarrow P \rightarrow M$ is often used, but not great) is a $G-$fibre bundle, with extra structure given by $G$ (as a manifold) as a right $G-$ torsor (treating $G$ as a lie group). It acts on itself by multiplying elements of itself on the right.
When treating $G$ as a Torsor, we forget about it's distinguished element.
def: a Torsor of a group is a space equipped with an action that is free and transitive.