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Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.

8
votes
Let $G$ be a Lie group and $H\subseteq G$ a closed subgroup. The quotient map $G\rightarrow G/H$ is a principal $H$-bundle. In particular, it is an example of a fibration. We then have an associated l …
answered Oct 1 '13 by Peter Crooks
4
votes
1answer
Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting representati …
asked Feb 16 '13 by Peter Crooks
3
votes
2answers
Suppose that $f:X\rightarrow Y$ is a homotopy equivalence of manifolds. Given a manifold $F$, the pullback construction for $f$ yields a correspondence between isomorphism classes of fibre bundles ove …
asked Feb 28 '13 by Peter Crooks
7
votes
The answer is that $\pi_1(G/G_x)=0$. Use the long-exact sequence of homotopy groups one obtains from a fibration. In this case, the fibration is $G_x\rightarrow G\rightarrow G/G_x$. We have $$\ldots\r …
answered Feb 21 '14 by Peter Crooks
10
votes
3answers
Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure …
asked Feb 13 '13 by Peter Crooks
4
votes
1answer
Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes calle …
asked Feb 19 '13 by Peter Crooks