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The loop space Ω_X of a pointed topological space X is the space of based maps from the circle \mathbb S^1 to X with the compact-open topology.
10
votes
3
answers
588
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On the naturality of the bar construction
Let X be a based space. Then the Moore loop space MX is defined to be the topological monoid whose points are based loops [0,a] \to X where a \ge 0 is allowed to vary. Composition is gotten b …
9
votes
Accepted
What is the delooping of a looping?
A simple example should indicate the general phenomenon: Let A be a discrete based set.
The \Omega A is a point, so B \Omega A is a point.
The general phenomenon is this:
B\Omega A is always …
8
votes
1
answer
1k
views
The free smooth path space on a manifold
Let M be a closed, smooth manifold and let PM be the space of unbased piecewise smooth paths [0,1] \to M. Then restricting a path to its boundary gives a map
PM \to M \times M .
Question …
5
votes
Proof of the ''trangression theorem''
They are equal up to sign.
If F\to E\to B is a Hurewicz fibration, where B is well-pointed, then we have a factorization E\to E/F \to B and we have the Barratt-Puppe extension $E/F \to \Sigma …
10
votes
classifying space of orthogonal groups
BO can be defined as the colimit over (k,n) of Grassmanians G_k(\Bbb R^n) of k-dimensional linear subspaces of \Bbb R^n (the limit over n is defined by standard inclusions $\Bbb R^n \subse …