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2
votes
2answers
189 views

Infinite loop of a p-completed specta vs p-completion of infinite loop of the spectra

Assume that we have a connective spectrum $X$, and denote the $p$-completion of this spectrum in the sense of Bousfield by $X^{\wedge}_p$ (which is given by the function spectrum ...
5
votes
1answer
84 views

Is there any relation between the simplicial $S^1$ and the Hochschild homology of a noncommutative algebras

Let $k$ be the base field and $A$ be a unital associative $k$-algebra. Let's review the Hochschild homology theory: we have the Hochschild chain comple $C_{\cdot}(A)$ where $$ C_n(A):=A^{\otimes n+1} ...
7
votes
3answers
408 views

Real varieties with enough algebraic loops

Let $(X,\sigma)$ be a complex variety with complex conjugation (equivalently, an algebraic variety over $\mathbb R$). We use the notations $X(\mathbb R):=X^\sigma$ for the set of fixed points of $X$ ...
1
vote
0answers
67 views

Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$

Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, ...
11
votes
4answers
864 views

Is there a good way to understand the free loop space of a sphere?

I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$. Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type. I ...
4
votes
3answers
351 views

Where does Segal's category come from?

Segal's category $\Gamma$ is the skeleton of the category $\text{FinSet}_{\ast}$ of pointed finite sets. It is used to write down $\Gamma$-spaces, which are functors $\Gamma \to \text{Top}$ satisfying ...
0
votes
0answers
84 views

loop space homology and lens spaces

Is the homology of free loop space of lens spaces known? Thanks in advance for your help.
2
votes
0answers
88 views

Chen's iterated integrals and free loop space

I recently found out about Chen's iterated integrals for paths in a differentiable manifold, and I was wondering if an analogous construction exists for free loops, i.e. a set of variables one ...
3
votes
0answers
179 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 structure of a monoid. ...
10
votes
2answers
409 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 $\Omega BX$ is homotopy ...
9
votes
3answers
331 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 ...
1
vote
2answers
433 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? Here, loop spaces ...
3
votes
1answer
297 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 homotopy equivalent ...
7
votes
1answer
360 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 simplicial set to a ...
15
votes
3answers
918 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 $\Lambda C$ whose nerve ...
8
votes
1answer
408 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 following "differentiable ...
4
votes
0answers
117 views

What is the grading of x(x−1)R[x]? Loopspace for Karoubi-Villamayor K-theory.

I am reading the chapter on Karoubi-Villamayor K-theory in Weibel's K-book. In particular he defines $\Omega R=(x^2-x)R[x]$ for a ring. This will eventually lead to a model for the loopspace ...
2
votes
2answers
322 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 that there exists ...
10
votes
3answers
505 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. $C_{n}(A,A)=A^{\otimes (n+1)}$). This is a ...
13
votes
1answer
461 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 $G$-manifold: the ...
5
votes
2answers
714 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 $M$). I see the free ...
6
votes
1answer
504 views

Haar measure on infinite dimensional Lie groups?

Hi. Is there a Haar measure or equivalent on infinite dimensional Lie groups? I've been playing around with $Diff(S^1)$, and at least a direct approach seems quite hopeless. It goes something like ...
6
votes
1answer
519 views

Homotopy type of the self-homotopy equivalences of a bouquet of spheres

Before I state the questions I have in mind, let me give some background. If one considers $S^2$ then it is known due to Kneser that $\textrm{Homeo}^{+}(S^2)$ has the homotopy type of $SO(3)$. By ...
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 abelian and acts ...
2
votes
1answer
253 views

First cohomology of the space of long knots in R^4

Let's consider the space of long knots in $\mathbb R^n, n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I ...
5
votes
1answer
343 views

What space does “Loop infinity of the infinite dimensional sphere” refer to?

I've been reading Hatcher's survey "A Short Exposition of the Madsen-Weiss Theorem". In it, he discusses the Barratt-Priddy-Quillen theorem, which says that the homology of the infinite symmetric ...
12
votes
1answer
450 views

What is known about the connection of positive energy representations of loop groups and modular forms

At the end of Section 14.1 in Pressley, Segal "Loop Groups" there is the remark that the partition function is a modular function in the sense that the Dedekind $\eta$ function is a modular form. I ...
6
votes
4answers
789 views

I was wondering if the set of singular loops is a (somewhere) submanifold of loop space?

The set of all smooth maps $S^1\to M^n$ ($M$ is a smooth manifold) is a generalized manifold(see http://ncatlab.org/nlab/show/smooth+loop+space). I was wondering if the set of singular loops (maps ...
18
votes
0answers
636 views

local equivalence of loop group representations

Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG ...
6
votes
2answers
707 views

Uniqueness of loop spaces

Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$. Under what assumptions is (the homotopy type of) $Y$ unique? As has been pointed out below, the ...
15
votes
3answers
703 views

central extensions of Diff(S^1) and of the semigroup of annuli

$\mathit{Diff}(S^1)$ refers to the group of orientation preserving diffeomorphisms of the circle. The semigroup of annuli $\mathcal A$ is its "complexification": the elements of $\mathcal A$ are ...
5
votes
1answer
504 views

Classifying spaces of E_1 - spaces

Hello, I try to understand aspects of homotopy coherence, in particular "recognition principle" of May. About the following I did not think a lot, but I decided to ask here anyway, so to save ...
4
votes
2answers
442 views

What are the algebras over $\Omega^k\Sigma^k$ ?

Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair $$ \Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k, $$ where $\Sigma^k$ is the $k$-th supension functor and ...
6
votes
6answers
487 views

Equivariant homology of $\Omega X$\/-space (references needed)?

Let $(X, *)$ be pointed a (1-connected) space, and let $\Omega X$ denote its based loops space. Then, as one knows very well, $\Omega X$ is a group up to homotopy (this includes all the necessary ...
1
vote
2answers
679 views

A question about the affine Grassmanian

For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as: $$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$ Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. ...
5
votes
2answers
615 views

Proof of the ''trangression theorem''

Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
1
vote
0answers
378 views

Recognition principle

Hello, The classical statement of the recognition principle (after Boardman, Vogt, Milgram and May) that I know is: Let $X$ be a (group-like) topological space acted on by the little $n$-discs ...
7
votes
2answers
341 views

Infinite loop space maps into or out of BAut(F_n)

There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become ...
0
votes
5answers
520 views

Loopspace of an Eilenberg Maclane space K(G,n)

I've seen the fact that the loopspace $\Omega K(G,n)$ is homotopy equivalent to $K(G,n-1)$ mentioned in some places, but I have no idea why. Can anyone offer a good explanation? Also, what happens ...
4
votes
1answer
279 views

What do representations of infinite-dimensional Heisenberg groups look like?

I'm interested in an infinite dim'l Heisenberg group associated to the vector space $V = L\mathbb{C}/\mathbb{C}$ = {$f \colon S^1 \to \mathbb{C}$|$f$ smooth}/(const. maps). The group is ...
1
vote
1answer
192 views

Commutativity of a diagram of boundary morphisms from the long exact sequence of homotopy groups of a fibration and its loop spaces

Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber. Question: How do you prove that the following diagram of homotopy groups commutes?: $\pi_n(Y) \to ...
2
votes
3answers
296 views

parameterizing polynomial loops in $\mathbb{C}^\times$

Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L_{poly}\mathbb{C}^\times$ of loops that look like $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ ...
17
votes
2answers
976 views

What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
9
votes
2answers
1k views

Explicit cocycle for the central extension of the algebraic loop group G(C((t))).

Let G be a simple Lie group and let G(ℂ((t))) be its loop group. The Lie algebra g[[t]][t-1] has a well known central extension (see e.g. Wikipedia) given by the cocycle c(f,g) = Res0 < f ...
11
votes
1answer
488 views

Are centrally extended p-adic groups defined over F_1?

Let G be a semisimple algebraic group. Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor G : Rings → Groups by the second algebraic ...
3
votes
0answers
227 views

Maps of loop spaces with infinity-bounded differential.

I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally: In the following piece-wise smooth means smooth on ...
29
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" algebra over the ...
24
votes
5answers
2k views

Topologists loops versus algebraists loops

Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also ...
20
votes
2answers
940 views

Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?

There are two ways to define smooth mapping spaces and I want to know how they compare? Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
18
votes
5answers
2k views

Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...