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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7
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1answer
405 views

Homotopy type of continuous/smooth/analytic loop spaces?

Apologies in advance if this is well-known; a google search did not produce anything useful. Let $(M,p)$ be a pointed real analytic manifold. Are the (free or pointed) loop spaces of continuous, ...
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2answers
281 views

Jones' theorem for non-simply-connected spaces?

Let $X$ be a smooth manifold. Jones' theorem says that $H^\bullet(\mathcal{L}X)\cong HH_\bullet(\Omega^\bullet_X)$, where $\mathcal{L}X$ is the free loop space of $X$. Is there a modification of this ...
6
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1answer
112 views

Is there a filtered splitting of product labelling spaces?

For a well-based space $X$ denote by $C(\mathbb{R};X)$ the unordered configuration space of points on the real line with labels in $X$, and a point can vanish if its label reaches the basepoint. (...
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281 views

Low-Dimensional Spaces with High-Dimensional Homology

Barratt-Milnor Spheres $X_n$ are spaces with finite topological dimension $n$ but which have non-vanishing singular homology in arbitrarily high dimensions. Here, they prove that if $n > 1$ then ...
5
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1answer
167 views

Integral homology of braid groups as a ring

Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each ...
5
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1answer
115 views

Homology of the free loop space of generalized flag varieties

Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{...
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1answer
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Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?

For a space1 $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space. Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X$...
19
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342 views

Todd class as an Euler class

Let $X$ be a relatively nice scheme or topological space. In various physics papers I've come accross, the Todd class $\text{Td}(T_X)$ is viewed as the Euler class of the normal bundle to $X\to LX$. ...
3
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1answer
150 views

$E_\infty$-space structure of $B\mathrm{GL}(\mathbb S_{(p)})$

In Geometric Topology - Localization, Periodicity, and Galois Symmetry by Dennis Sullivan, we can read that there is a decomposition $$B\mathrm{SL}(\mathbb S_{(p)})\times K((\mathbf Z_{(p)})^\times)\...
6
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1answer
208 views

Relative homology of free loop space with respect to constant loops

Let $Q$ be a closed manifold with $\dim Q\geq2$ and let $\Lambda_0Q$ be the connected component of the free loop space of $Q$ whose elements are contractible loops. I am looking for conditions on the ...
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281 views

It is possible that $ X \simeq ΩX $? and that $ X \simeq Ω^ 2X $?

Original post: https://math.stackexchange.com/questions/3810423/it-is-possible-that-x-simeq-%ce%a9x-and-that-x-simeq-%ce%a9-2x I am studying J. Strom's Modern Classical Homotopy Theory. In chapter 4 ...
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A question on recognition of equivariant loop spaces

I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places. We know from the work of Segal that to give a loop ...
8
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2answers
554 views

Why study infinite loop spaces?

What makes an infinite loop space an interesting object of study for homotopy theorists? The reason I ask this question is that I found a lot of results treating the question of whether a given space ...
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Equivariant splitting of loop space of a suspension

It is well known, e.g. by Cohen's "A model for the free loop space of a suspension", that there is a stable splitting of the free loop space $\mathcal{L} \Sigma X $of the suspension $\...
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Group cohomology as homotopy groups

Let $G$ be a group and $A$ a group with a $G$-action. Then in general, $H^0(G;A)=A^G$ is a group, and $H^1(G;A)$ is simply a pointed set. If $A$ is an abelian group, then $H^i(G;A)$ exists and is an ...
8
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1answer
518 views

Loop spaces motivation

I read that one of the main goals of utilization simplicial methods is to prove that a space is a loop space. On the other hand where lies the main importance to recognize topological spaces as loop ...
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Comparison of two well-known bases of the integral homology group of based loop group

Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways: (1) Via Bott-Samelson'...
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168 views

Where can I find W. Browder's thesis

I've been looking for W. Browder's thesis Homology of loop spaces for a while now, and I really found nothing except for articles and book having it in their bibliography. Does someone know if it can ...
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Grothendieck Riemann Roch is abelian localisation on loop spaces

Abelian localisation says approximately that for a proper equivaraint map $f:X\to Y$ between schemes with a $\mathbf{G}_m$ action, the pushforward on cohomology $f_*\omega$ can be computed by the ...
8
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1answer
621 views

For which G is BLG weak homotopy equivalent to LBG?

Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?)...
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1answer
205 views

Infinite loop space of ring spectra: the cup product

I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
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1answer
468 views

If $\Omega X \simeq \Omega Y$ then is $X \simeq Y$ for $X,Y$ simply connected?

If $\Omega X \simeq \Omega Y$ then is $X \simeq Y$ for $X,Y$ simply connected? Assuming $X,Y$ are nice spaces like CW of course. Clearly this is true by Whitehead, but I am looking for a more ...
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1answer
876 views

The free smooth path space on a manifold

Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map $$ PM \to M \times M . $$ Question ...
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1answer
229 views

Delooping the quotient space $SU/SU(n)$

Let $SU$ denote the infinite unitary group. Does the quotient space $SU/SU(n)$ admit a delooping $X$? One could also ask that this space $X$ sit in a fiber sequence $BSU(n)\to BSU\to X$, but this is ...
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2answers
339 views

Sheafification of loop scheme/group

Let $X$ be a scheme over $K = k((t))$, where $k$ is a field. We define the loop scheme $LX$ to be the functor from the category of $k$-algebras to sets by $R \mapsto LX(R) := X(Spec (R((t))))$. Do we ...
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512 views

How are characteristic classes morphisms of infinite loop spaces? (if they are)

The direct sum of real vector bundles endows $BO=\mathrm{colim} BO(n)$ with a natural structure of abelian group up to homotopy. The same applies to the classifying spaces of all groups in the ...
8
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1answer
242 views

When does $BG \to BA$ loop to a homomorphism?

If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) ...
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1answer
937 views

The free loop space of spheres

Let $n>1$. The homology of the free loop space $\Lambda S^n$ of the sphere $S^n$ contains two torsion if $n$ is even. Thus the fibration $$ \Omega S^n\rightarrow \Lambda S^n\rightarrow S^n $$ is ...
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Exterior derivative on loop space

Notations: Let $X$ be a manifold, and denote by $LX := C^\infty(S^1,X)$ its loop space. For a loop $\gamma \in LX$ we can think at the tangent space of $LX$ at the point $\gamma$ as the space of ...
15
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1answer
526 views

Mapping a loop space to quaternionic projective space

Let $\mathbf{H}P^\infty$ denote the infinite-dimensional quaternionic projective space. The inclusion of its bottom cell defines a map $S^4 \to \mathbf{H}P^\infty$. Does this extend to a map $\Omega S^...
9
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1answer
497 views

Homology of the free loop space of a Grassmanian

Is there any reference for calculation of the rational homology of the free loop space $H_*(\mathcal{L}Gr(k,n),\mathbb{Q})$ of a complex Grassmanian? More precisely, I am interested in computing ranks ...
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97 views

The tangent space to the Hilbert manifold of $H^1$ loops at non-smooth loops

It is satisfactory to have a nice functional analytic setting for the energy functional in Riemannian geometry. I'm currently deep into Klingenberg's book "Riemannian geometry" which (among other ...
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81 views

Unstable and stable looping and delooping

I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an ...
9
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1answer
407 views

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 $\psi^\ell-1$ for $\ell$ a generator ...
6
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1answer
448 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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2answers
354 views

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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1answer
654 views

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
446 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^...
18
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1answer
470 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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177 views

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 ...
7
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1answer
213 views

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}...
6
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2answers
476 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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201 views

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 ...
7
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0answers
211 views

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 \...
7
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0answers
205 views

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 $\Omega$...
5
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1answer
313 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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2answers
417 views

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 ...
3
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1answer
100 views

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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237 views

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 ...
11
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376 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 ...