9
votes
2answers
304 views
Proving that a space cannot be delooped.
Suppose we have some pointed connected topological space $X$. How can we determine if there exists a space $BX$, called delooping of $X$, such that its space of based loops $\Omega …
2
votes
0answers
135 views
looping and delooping spaces and categories
I'm trying to understand the relationship between the notions of looping and delooping in category theory and topology.
The morphisms in a category with one object have the struct …
8
votes
3answers
283 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. Com …
1
vote
2answers
299 views
Are loop spaces of homotopically equivalent spaces homotopically equivalent? [closed]
Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces.
Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence …
14
votes
3answers
868 views
Loop space of a category
This seems like it should be a "standard" thing, and I think I remember even seeing it somewhere, but I can't remember where.
Let $C$ be a small category. Is there a category $\L …
3
votes
1answer
242 views
Free Loops, Moore Paths and the Borel Construction
My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homo …
7
votes
1answer
232 views
Is there an algebraic “derived mapping space” construction that encompasses both Hochschild homology and loop spaces of non-simply-connected spaces?
I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a …
8
votes
1answer
370 views
Vector fields on path spaces
I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the followin …
10
votes
3answers
459 views
Geometric realization of Hochschild complex
Let $A$ be a commutative $\mathbb{C}$-algebra, and consider $C_{\bullet}(A,A)$ the simplicial Hochschild homology module of $A$ with respect to itself (i.e. $C_{n}(A,A)=A^{\otimes …
10
votes
5answers
1k views
Homology of loop space
I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available here).
On page 8, he states the following theorem. Let $X$ be a space such that $\pi_1(X)$ is …
27
votes
3answers
2k views
Algebras over the little disks operad
Hello,
The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied.
My problem is the following:
The "recognition principle" says that every "group-like …
5
votes
2answers
546 views
The connected components of the free loop space
I am trying to understand the topology (in terms of homology groups) of the free loop space $\Lambda M$ of nice spaces (Complete Riemannian connected finite dimensional manifolds $ …
13
votes
1answer
415 views
free loop space and invariant forms
Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact …
16
votes
0answers
544 views
local equivalence of loop group representations
Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let $LG:=C^\infty(S^1,G)$ be its smooth loop group. Given an interval $I\subset S^1$, we have the l …
2
votes
2answers
274 views
2-cocycle on LSU(2)
$SU(2)$ is a Lie group, with a Lie algebra $\mathfrak{su}(2)$. I now consider the loop group
$ LSU(2) = \{ \gamma: S^1 \to SU(2); \gamma \mathrm{ smooth} \} $. It is well-known th …

