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Results tagged with loop-spaces
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user 318
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.
15
votes
Accepted
Can the standard map $\Sigma \Omega X \to X$ be a homotopy equivalence?
If $X$ is such a space (a CW complex, say), then it must be a suspension $\Sigma Y$ (as it is the suspension of $y :=\Omega X$). The James splitting gives
$$\Sigma \Omega \Sigma Y \simeq \bigvee_{n=1} …
- 18.2k
5
votes
On the naturality of the bar construction
I think the following will do, but it is not pretty.
Write an element of $\Delta^n$ as a tuple $0 \leq x_1 \leq \cdots \leq x_n \leq 1$. Identifying $[0,1]$ with $[-\infty, \infty]$, we may as well c …
- 18.2k
9
votes
Accepted
Homology of loop space
I believe the argument Galatius had in mind is the following. Let us write $\bar{V} = \oplus_{n \geq 2} V_n$, so $V = V_1 \oplus \bar{V}$. All cohomology will be rational, and we write $G := \pi_1(X)$ …
- 18.2k
7
votes
Infinite loop space maps into or out of BAut(F_n)
In the following paper
U. Tillmann, The representation of the mapping class group of a surface on its fundamental group in stable homology, Q J Math (2010) 61 (3): 373-380.
Ulrike Tillmann studies t …
- 18.2k