How does one calculate homotopy classes for group coset spaces?

Question

Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times SU(3)_R)/SU(3)_{\rm diag}\cong SU(3)$ and so the coset space is itself a group, but how does this extend to more general examples like say $G/H=SU(5)/(SU(3)\times SU(2)\times U(1))$?

What about the case where the groups are non-compact, say they're spacetime symmetry groups? For example, $G/H= ISO(4,1)/ISO(3,1)$ or $G/H=SO(4,2)/SO(3,1)$?

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user homotopyquestions

Comments

I think this question is not complete enough. In particular, it is not stated what homotopy classes of maps you want to compute.

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user Fernando Muro

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Witten computes $\pi_4(SU(3))$ and $\pi_5(SU(3))$ in his paper so I suspect that the OP wishes to know how to calculate the homotopy groups of group coset spaces.

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user j.c.

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Noncompact Lie groups are homotopic to their maximal compact subgroups, so you can reduce to those.

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user Allen Knutson

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I'm pursuing a crusade so that people do not confuse being homotopic, which is a relation among maps, with being homotopy equivalent, a relation for spaces

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user Fernando Muro

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I don't think you need a crusade for this; the question is simply ambiguous, and the OP should state more clearly what is being asked for: some homotopy groups of these spaces, a homotopy classification of these spaces, or perhaps something entirely different.

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user Danny Ruberman

Answer 1

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 long-exact sequence of homotopy groups,

$$\ldots\rightarrow\pi_n(H)\rightarrow\pi_n(G)\rightarrow\pi_n(G/H)\rightarrow\pi_{n-1}(H)\rightarrow\ldots.$$ So, if you have information about the homotopy groups of $G$ and $H$, you might be able to obtain information about those of $G/H$ using this sequence.

This post imported from StackExchange MathOverflow at 2015-03-15 10:02 (UTC), posted by SE-user Peter Crooks