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Questions tagged [loop-spaces]

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.

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Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?

Quillen famously constructed a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^q-1$. Does this lift to the ...
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1answer
366 views

Unstable Greek letter elements

A theorem of Hopkins and Mahowald states that the Thom spectrum of the map $\Omega^2 S^3 \to B\mathrm{GL}_1(\mathbb{S}_{(p)})$ classifying the element $p$ is exactly $\mathrm{H}\mathbf{F}_p$. Let $T(1)...
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Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
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CW complex of iterated loop spaces

In Milnor's book Morse Theory, it is proved that the loop space $\Omega S^n$ of the n sphere has the homotopy type of a CW complex with one cell each in the dimensions 0, n-1, 2n-2, 3n-3, ... Or more ...
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1answer
354 views

Homotopy orbits, spectra and infinite loop spaces

Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results, $$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$ and $$\Omega^...
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1answer
397 views

Is $\Omega J_{p^n-1}S^2$ commutative up to homotopy?

Fix a prime $p\geq 5$ and an integer $n>0$. All spaces in this question are implicitly $p$-localized. Consider the spaces $X=J_{p^n-1}S^2$ (the $p^n-1$'th stage in the James construction $JS^2\...
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Loop space in Topological sense v.s. Categorical sense

I know that the loop space of given pointed topological space $(X,\ast)$ is the set of pointed maps $\mathrm{Map}_\ast(S^1,X)$. I would denote it by $\Omega X$. On the other hand, I saw an article in ...
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1answer
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Is the homology of $\Omega^2\Sigma^2X$ free as a Gerstenhaber algebra?

Let $X$ be a connected space. According to Getzler BV-algebras and two-dimensional topologcial field theories , page 271, we have and isomorphism $ H_*(\Omega^2\Sigma^2X) \cong {\cal G}( \widetilde{H}...
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2answers
328 views

Integer homology of double loop space of odd-dimensional sphere

I have checked everything "homology of loop spaces"-like, but was not able to find what is $H_*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?
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The recognition principle and CGWH spaces

The recognition principle [Boardmann–Vogt, May] states that a grouplike algebra over the little $n$-disks/cube operad is weakly equivalent to an $n$-fold loop space. There are technical hypotheses ...
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Loop space functor and sequential colimits of inclusions

The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is: Let $X_0\hookrightarrow X_1 \...
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Notation: Why Ω for the based loop functor?

This is just a question about notation - probably useless, but it's always baffled me: Why was $\Omega$ chosen to denote the based loop functor? I once heard someone speculate: "It's because $\Omega$...
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1answer
251 views

Stable homotopy groups of $QX$

If $X$ is a space, we can form $QX=\varinjlim \Omega^n\Sigma^nX$ which is an infinite loop space with homotopy groups $\pi_i(QX)=\pi^{s}_i(X)$ the stable homotopy groups of $X.$ But these are the ...
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The homotopy pullback of a point along itself is the loop space

I have seen on the nLab that we can view the loop space as a particular homotopy pullback, and that it is even the way a "loop space object" is defined in general (when it exists). Can someone give ...
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1answer
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Adjoint map of $\Gamma$-space prespectrum

I am looking for a reader-friendly proof of the following theorem: let $A$ be a special $\Gamma$-space then $\pi_0(A(S^0))$ is a commutative monoid (I have proved up to this), if further it is an ...
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What is known about maps between loop spaces of Spheres? - Reference request

What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
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336 views

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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167 views

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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107 views

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 $T^1S^n$, ...
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1answer
129 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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3answers
707 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 ...
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1answer
357 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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0answers
118 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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1answer
233 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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2answers
364 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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1answer
125 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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1answer
151 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 ...
7
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1answer
441 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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1answer
266 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 ...
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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 $T(M)$...
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1answer
82 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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1answer
221 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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1answer
202 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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1answer
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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 $$ \eta_n=\phi^{-...
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1answer
709 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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1answer
149 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 _{k=1}^\...
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1answer
137 views

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 $\Omega\...
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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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2answers
327 views

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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2answers
462 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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1answer
343 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 $\mathcal{C}...
7
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1answer
220 views

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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1answer
673 views

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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1answer
566 views

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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1answer
154 views

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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167 views

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 $S^1\to X$ ...