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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 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 …
John Klein's user avatar
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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 …
John Klein's user avatar
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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 …
John Klein's user avatar
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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 …
John Klein's user avatar
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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 …
John Klein's user avatar
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