The loop-spaces tag has no usage guidance.

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### Geometric Satake and Restriction

The Geometric Satake correspondence (due to Lusztig, Ginzburg, Mirkovic-Vilonen) relates perverse sheaves on the Loop Group $\hat{G}$ (with their convolution product) to the Representations of the ...

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### Eilenberg-Moore spectral sequence for path-loop fibration over Q\Sigma X (reference request)

Related to the question here, here is another question. Consider the kernel of the map $H_*(QY;Z/p)\rightarrow H_{*+1}(Q\Sigma Y;Z/p)$. restricted to
$PH_*(QY)$, and let's say $Y$ itself is a ...

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### The behaviour of the suspension homomorphism on $H_*(QX;Z/p)$ for odd $p$ (Reference request)

The mod $p$ homology of $QX=\Omega ^{\infty}\Sigma ^{\infty}X$ for connected $X$ was computed by Dyer-Lashof Homology of Iterated Loop Spaces, Amer. J. of Math., vol.84, No.1 pp 35-88. 1962 It follows ...

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### Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...

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### Differentiability of a map to the free loop space

While reading Morse theory, closed geodesics, and the homology
of free loop spaces, the author claims the following:
Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...

**3**

votes

**1**answer

104 views

### Does Kähler structure on X imply Kähler structure on the loop space of X?

Does Kähler structure on $X$ imply Kähler structure on the loop space ($LX$) of $X$? Since the loop space of $X$ is the space of maps from the circle $S^1$ to $X$, I suspect one may use the pullback ...

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407 views

### classifying space of orthogonal groups

Let $O(n)$ be the $n$-th orthogonal group and $O$ be the direct limit of $O(n)$ with respect to $n$. Let $BO(n)$ and $BO$ be the classifying spaces.
Question:
Why $BO$ is an $H$-space? My supervisor ...

**3**

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**1**answer

212 views

### the “Kahn-Priddy map” and “multiplicative $p$-local equivalence”

The following is a part of a paper that I need to understand
I totally do not know the argument. Could you explain? Thanks.
Let $\Sigma_n$ be the $n$-th symmetric group and $\Sigma_\infty$ be the ...

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87 views

### Double loop groups and cohomology

Let $G$ be a connected reductive group over $\mathbb{C}$ of Lie algebra $\mathfrak{g}$.
What is the value of $H^{3}(\mathfrak{g}((t))((s)),\mathbb{C})$?

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216 views

### Can the standard map $\Sigma \Omega X \to X$ be a homotopy equivalence?

The question is in the title : are there spaces X such that the adjoint of the identity on the loop space $\Omega X$, i.e. $\Sigma\Omega X \to X$, is a homotopy equivalence ?

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299 views

### Seifert--van Kampen for the loop space dga

I recently noticed that I could mostly prove a special case of the following statement. I think it's true in general, though perhaps only for nice spaces.
Let $X$ be a topological space, and ...

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vote

**1**answer

94 views

### dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32
Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...

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votes

**1**answer

131 views

### cohomology ring of infinite iterated loop space

What is the cohomology ring
$$
H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)?
$$
I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...

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votes

**1**answer

172 views

### cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra
(1)
$$
H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...

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votes

**1**answer

240 views

### coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...

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### cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring
$$
H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p)
$$
is obtained for any ...

**0**

votes

**0**answers

459 views

### maps from labelled configuration space to section space / iterated loop space

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3:
for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle ...

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**1**answer

60 views

### Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology
$$
\alpha_n: C(\mathbb{R}^n;X)\to ...

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148 views

### configuration space and iterated loop space

Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...

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**1**answer

187 views

### Direct proof that $U$ is an $E_\infty$-space

An immediate consequence of Bott periodicity is that the infinite unitary space is an infinite loop space and so an $E_\infty$-space. I wonder if there is a direct proof (not using $U = \Omega^2 U$) ...

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**1**answer

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### iterated loop spaces and configuration spaces [closed]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
...

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354 views

### When does the free loop space fibration split?

This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be ...

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**1**answer

93 views

### stable splitting into a wedge sum [closed]

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee _{k=1}^\infty Y_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee ...

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### Loop space of Fredholm operators from a Relative loop space

Atiyah and Singer proved that the nontrivial component of the set of skew-adjoint Fredholm operators $ \hat{\mathcal{F}_{*}}(\mathscr{H})$ is homotopic to the loop space of Fredholm operators ...

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### Symplectic structures on the grassmannian model of the based loop group

$\newcommand{\Ad}{\operatorname{Ad}}$
In the study of (smooth/algebraic) based loop spaces of compact groups, one often uses a Grassmannian model to study the space. In particular, the Grassmannian ...

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### Example s.t. the unbased loop-space is not $\Omega X \times X$

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...

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374 views

### Maps to the group completion

Let $M$ be an H-space, topological monoid (homotopy-commutative if necessary):
What does the group comletion $\Omega BM$ represent in homotopy category? Is $[X,\Omega B M]$ always equal to the ...

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**1**answer

245 views

### Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces

In Geometry of the iterated loop space, Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.
There exist $\Sigma$-free operads ...

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**1**answer

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### Closed geodesics in free smooth loop space?

I know very little about these subjects, so I apologise if this is a naive line of inquiry:
Let $M$ be a smooth $n$-dimensional Riemannian manifold. I understand that it is possible to construct an ...

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### Free Loop-Space Recognition Principle

It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...

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### What are iterated cobar constructions?

In Beck's paper "On H-spaces and Infinite Loop Spaces", he states that every algebra over the monad $\Omega^k$$\Sigma^k$ is a $k$-fold loop space. He proves the trivial case k = 0 when this is the ...

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### Homologically distinct infinite loop structures on a space

Let $X$ be a connected pointed topological space equipped with two different actions of $E_\infty$-operad. Each action provides a collection of deloopings $X_i$, where $X_0 = X$ and $\Omega X_i$ is ...

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### cohomology algebra of unordered configuration space on Euclidean space

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents):
Line 2: ... is used to compute the precise algebra ...

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### Topological characterisation of loop spaces

Let $\Omega\colon \mathrm{Top}_*\to\mathrm{Top}_*$ be the loop space functor assigning to each pointed topological space $X$ the pointed space consisting of all based continuous maps $S^1\to X$ ...

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### Which homology classes from loop space?

Fix a closed connected manifold $Q$ and let $LQ$ denote its free loop space. We can get second homology classes on $Q$ by "doing things" to loops in $Q$.
For instance, if we have a loop of loops, it ...

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### $E_{\infty}$ spaces are $A_{\infty}$ spaces

While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, ...

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373 views

### Reconciling the affine grassmannian and the based loop group

I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...

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vote

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200 views

### The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?

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### Reference request: modern proof of the recognition principle [closed]

I'm looking for a modern version of P. May's proof of the recognition principle (in "geometry of iterated loop spaces"). Can someone help me?

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### Is this groupoid a model for the derived fixed-point locus of the free loop space?

In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...

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388 views

### What is the delooping of a looping?

What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$?
The ...

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**1**answer

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### Advantage in Using Cyclic Homology to a compute Equivariant (Co)Homology of Loop Spaces

I am trying to compute equivariant (co)homology of the free loop space of a manifold $M$ that is not a Lie group, $H^{S^1}_*(LM)$ with the natural rotation action of $S^1$ on the loops of the free ...

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### 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 ...

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### 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}
...

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### 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 $X$ ...

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### 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 $G$, ...

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### 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 ...

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### 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 ...

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### loop space homology and lens spaces

Is the homology of free loop space of lens spaces known?
Thanks in advance for your help.

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### 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 ...