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Results tagged with loop-spaces
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user 8032
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
views
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k