4 correction again, was screwed up between symmetric elements in the tensor square and bilinear pairings

This, in general, an incredibly difficult problem. Even we just want to compute the rational homotopy groups of the suspension of X and X is simply connected, where we can do everything using rational homotopy theory to reduce things to commutative DGAs, this starts to involve things like free Lie algebras and the like.

Locally at the prime 2, there is actually a famous long exact sequence when X is a sphere called the EHP long exact sequence. It relates the suspension homomorphism, the Hopf map, and a "whitehead product" map. This gives rise to the EHP spectral sequence that, funnily enough, starts with the 2-local homotopy groups of odd-dimensional spheres and computes the 2-local homotopy groups of spheres. Miller and Ravenel have a paper titled "Mark Mahowald's work on the homotopy groups of spheres" that covers some of this material in detail.

Another approach is to say: The "stable" homotopy groups of X are a first-order approximation using the Freudenthal suspension theorem that Andrea mentioned. There is then a "quadratic" correction term that you can try to use to get an approximation of the homotopy groups that is correct out to roughly three times the connectivity, and so on. These lead into the subject of Goodwillie calculus.

For a classifying space K(G,1), neither of these approaches work very well, because the higher homotopy groups are going to depend pretty intricately on your group itself. For instance, pi3 of the suspension of the classifying space of a free group is the set of Z-valued symmetric bilinear pairings on elements in Gab Gab where Gab is the abelianization of G, and for a general group it lives in an exact sequence between something involving such pairings symmetric elements and the second group homology of G. I don't know a closed form for it but maybe someone else knows better.

EDIT: Let me at least be precise, there's an exact sequence

pi4 (Σ BG) -> H3 G -> Bil(G(Gab ⊗ Gab)Z/2 -> pi3 (Σ BG) -> H2G -> 0

where Bil(G) is the set .

Note that for R a ring, an element of Z-valued (Gab Gab)Z/2 gives rise to an R-valued symmetric bilinear pairings pairing on the abelianization of GHom(Gab, R).

EDIT AGAINFOR THE FINAL TIME: got myself confused and switched a cohomology computation sorry for a the multiple revisions, switching back and forth between homology oneand cohomology gave me errors. It's fixed the exact sequence above should be correct now.

3 corrected universal bilinear form vs set of bilinear pairings.

This, in general, an incredibly difficult problem. Even we just want to compute the rational homotopy groups of the suspension of X and X is simply connected, where we can do everything using rational homotopy theory to reduce things to commutative DGAs, this starts to involve things like free Lie algebras and the like.

Locally at the prime 2, there is actually a famous long exact sequence when X is a sphere called the EHP long exact sequence. It relates the suspension homomorphism, the Hopf map, and a "whitehead product" map. This gives rise to the EHP spectral sequence that, funnily enough, starts with the 2-local homotopy groups of odd-dimensional spheres and computes the 2-local homotopy groups of spheres. Miller and Ravenel have a paper titled "Mark Mahowald's work on the homotopy groups of spheres" that covers some of this material in detail.

Another approach is to say: The "stable" homotopy groups of X are a first-order approximation using the Freudenthal suspension theorem that Andrea mentioned. There is then a "quadratic" correction term that you can try to use to get an approximation of the homotopy groups that is correct out to roughly three times the connectivity, and so on. These lead into the subject of Goodwillie calculus.

For a classifying space K(G,1), neither of these approaches work very well, because the higher homotopy groups are going to depend pretty intricately on your group itself. For instance, pi3 of the suspension of the classifying space of a free group is something like the universal set of Z-valued symmetric bilinear pairing pairings on the abelianization, and for a general group it lives in an exact sequence between something involving such pairings and the second group homology of G. I don't know a closed form for it but maybe someone else knows better.

EDIT: Let me at least be precise, there's an exact sequence

H3 G -> Bil(G) -> pi3 (Σ BG) -> H2G -> 0

where Bil(G) is the universal module accepting a set of Z-valued symmetric bilinear pairing from pairings on the abelianization of G.

EDIT AGAIN: got myself confused and switched a cohomology computation for a homology one. It's fixed now.

2 state exact sequence precisely.

This, in general, an incredibly difficult problem. Even we just want to compute the rational homotopy groups of the suspension of X and X is simply connected, where we can do everything using rational homotopy theory to reduce things to commutative DGAs, this starts to involve things like free Lie algebras and the like.

Locally at the prime 2, there is actually a famous long exact sequence when X is a sphere called the EHP long exact sequence. It relates the suspension homomorphism, the Hopf map, and a "whitehead product" map. This gives rise to the EHP spectral sequence that, funnily enough, starts with the 2-local homotopy groups of odd-dimensional spheres and computes the 2-local homotopy groups of spheres. Miller and Ravenel have a paper titled "Mark Mahowald's work on the homotopy groups of spheres" that covers some of this material in detail.

Another approach is to say: The "stable" homotopy groups of X are a first-order approximation using the Freudenthal suspension theorem that Andrea mentioned. There is then a "quadratic" correction term that you can try to use to get an approximation of the homotopy groups that is correct out to roughly three times the connectivity, and so on. These lead into the subject of Goodwillie calculus.

For a classifying space K(G,1), neither of these approaches work very well, because the higher homotopy groups are going to depend pretty intricately on your group itself. For instance, pi3 of the suspension of the classifying space of a free group is something like the collection of universal symmetric bilinear pairings pairing on the abelianization, and for a general group it lives in an exact sequence between something involving such pairings and the second group homology of G. I don't know a closed form for it but maybe someone else knows better.

EDIT: Let me at least be precise, there's an exact sequence

H3 G -> Bil(G) -> pi3 (Σ BG) -> H2G -> 0

where Bil(G) is the universal module accepting a bilinear pairing from the abelianization of G.

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