Relationship betwen homotopy groups?

[quasar987]

Is there a relationship between the homotopy groups of a pair (X,A) and of the quotient X/A ? It feels like they should be equal under mild hypothesis.

More precisely, I am interested in the case where X is a smooth manifold and A a submanifold.

Thx

 

[owlpride]

I don't know much homotopy theory, but your intuition is correct in homology: The homology of X/A is isomorphic to the homology of the pair (X,A) provided there is an open neighborhood of A in X that deformation retracts to A.

I apologize for not answering your question if you already knew that.

 

[Jamma]

By homotopy of a pair, do you mean homotopy classes of maps of cubes, or disks, which have their boundaries contained in A?

I could imagine that your intuition is correct, given certain conditions.

I did notice that if $$p :E \rightarrow B$$ is a fibration with fibre F, then
$$p_* : \pi_n (E,F,f_0 ) \rightarrow \pi_n (B,b_0)$$

is an isomorphism. So maybe you just need that the quotient map is a fibration, which it is in your case isn't it?

 

[quasar987]

I don't know. Is it? It looks that way though.

 

[Jamma]

Actually, I'm not 100%, I think that it will probably vary very delicately on the submanifold.

I know that if you have a closed subgroup (H, say) of a Lie group (G, say) then H is a submanifold of G and G-->G/H is a fibre bundle (and hence a fibration).

So this obviously wouldn't have been formulated if it worked for all submanifolds.

I guess that this won't be too helpful then, unless your submanfold sits "nicely" inside your other manifold

 

[Jamma]

Maybe this is helpful for you, from the wiki page for fibre bundles:

Quotient spaces

If G is a topological group and H is a closed subgroup, then under some circumstances, the quotient space G/H together with the quotient map π : G → G/H is a fiber bundle, whose fiber is the topological space H. A necessary and sufficient condition for (G,G/H,π,H) to form a fiber bundle is that the mapping π admit local cross-sections (Steenrod & 1951 §7).

The most general conditions under which the quotient map will admit local cross-sections are not known, although if G is a Lie group and H a closed subgroup (and thus a Lie subgroup by Cartan's theorem), then the quotient map is a fiber bundle. One example of this is the Hopf fibration, S3 → S2 which is a fiber bundle over the sphere S2 whose total space is S3. From the perspective of Lie groups, S3 can be identified with the special unitary group SU(2). The abelian subgroup of diagonal matrices is isomorphic to the circle group U(1), and the quotient SU(2)/U(1) is diffeomorphic to the sphere.

More generally, if G is any topological group and H a closed subgroup which also happens to be a Lie group, then G → G/H is a fiber bundle.

 

[Jamma]

Sorry, ignore everything I am saying, I am being an idiot.

The fibre map for Lie groups makes more identifications than the one you gave, it is the coset space.

Sorry about that ^^

 

[Jamma]

Just as a quick counter example, X could be the sphere, A the sphere as well, so that the homotopy groups are unchanged for (X,A) but X/A is the single point set, maybe this is more helpful :D

 

[quasar987]

Heh, thanks Jamma.