2
votes
0answers
39 views

Clarification about a double delooped H-space.

I've just started reading J.P. May's book The Geometry of Iterated Loop Spaces and am misunderstanding something. Somewhere, it's asserted that if an H-space X can be delooped twice, the its ...
4
votes
2answers
72 views

Derivation and meaning of a long exact sequence of a Homotopy Groups for pairs of spaces

I read that there are long exact sequences of homotopy groups for each pair of pointed spaces $(X,A,x_{0})$. Now I know that for an exact sequence that, as the example below denotes $f \text{ and } ...
2
votes
0answers
216 views

Action of the Fundamental Group on Higher Homotopy Groups.

First: here are a couple links of which I am looking at. I try to add the relevant information (at least to my understanding) from them. ...
2
votes
0answers
59 views

Γ-spaces and operads

I'm looking for a comprehensible reference that explains how $\Gamma$-spaces are related to $E_{\infty}$-operads. I've found some old publications but was hoping there are better references out there. ...
5
votes
1answer
164 views

Why is the recognition principle important?

The recognition principle basically states that (under some conditions) a topological space $X$ has the weak homotopy type of some $\Omega^k Y$ iff it is an $E_k$-algebra (ie. an algebra over the ...
5
votes
3answers
599 views

Showing that the loopspace $\Omega S^{\infty}$ is homotopic to $S^{\infty}$.

Showing that the infinite dimensional sphere $S^{\infty}$ is contractible is rather easy by constructing an explicit contraction (Hatcher gives a nice one). I thought it might be a nice exercise to ...