(see also Chern-Weil theory, parameterized homotopy theory)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
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Homotopy groups
Basic facts
Theorems
Give a topological ground field $k$ and a base topological space $B$, a short exact sequence of topological $k$-vector bundles over $B$ is a sequence of topological $k$-vector bundle homomorphisms over $B$ (i.e. continuous functions which are fiber-wise $k$-linear maps)
(where $0$ denotes the rank-zero bundle) such that $p$ is a surjection and $i = ker_B(p)$ in the injection of its fiber-wise kernel, hence such that over each point $b \colon \ast \overset{}{\longrightarrow} B$ we have a short exact sequence of $k$-vector spaces:
(over paracompact topological spaces short exact sequences of real vector bundles split)
If
the ground field is the real numbers $k = \mathbb{R}$,
the base space $B$ is a paracompact Hausdorff space,
then every short exact sequence of topological vector bundles (1) splits and exhibits the middle item as the direct sum of vector bundles, over $B$, of the left and the right item:
Sketch: Under the assumption on $B$, there exists (by this Prop.) a fiberwise inner product on $\mathcal{V}$. With this the splitting follows by th usual splitting of short exact sequences of real vector spaces, applied fiberwise: $\mathcal{V}_R$ is fiberwise identified with the orthogonal complement of $\mathcal{V}_L$.
Textbook accounts:
Lecture notes:
Last revised on August 16, 2021 at 11:23:45. See the history of this page for a list of all contributions to it.