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.
154
questions
14
votes
1
answer
1k
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 ...
- 1,610
11
votes
1
answer
631
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^...
- 751
18
votes
1
answer
534
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\...
- 55k
1
vote
0
answers
210
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 ...
- 59
8
votes
1
answer
282
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}...
- 1,935
8
votes
3
answers
865
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?
- 1,129
5
votes
0
answers
224
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 ...
- 4,850
8
votes
0
answers
280
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 \...
- 91
7
votes
0
answers
213
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$...
- 113
5
votes
1
answer
432
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 ...
- 1,991
3
votes
2
answers
738
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 ...
- 2,129
4
votes
1
answer
114
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 ...
- 447
8
votes
0
answers
247
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 ...
- 5,016
11
votes
0
answers
449
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 ...
- 1,063
3
votes
0
answers
227
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 ...
- 3,041
5
votes
0
answers
316
views
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 ...
- 3,041
4
votes
0
answers
319
views
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 ...
- 4,850
3
votes
0
answers
120
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$, ...
- 401
3
votes
1
answer
155
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 ...
- 111
6
votes
3
answers
2k
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 ...
- 1,970
3
votes
1
answer
522
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 ...
- 1,970
3
votes
0
answers
168
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})$?
- 3,432
6
votes
1
answer
337
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 ?
- 301
6
votes
2
answers
542
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 ...
- 787
1
vote
1
answer
156
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 ...
- 2,213
3
votes
1
answer
173
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)?
$$
I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...
- 2,213
8
votes
1
answer
898
views
cohomology of iterated loop space on spheres
In the book The homology of iterated loop spaces, the homology Hopf algebra
(1)
$$
H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...
- 2,213
2
votes
1
answer
371
views
coproduct of the homology of iterated loop space on spheres
Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...
- 2,213
2
votes
0
answers
195
views
cohomology ring of mapping spaces
In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring
$$
H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p)
$$
is obtained for any ...
- 2,213
1
vote
0
answers
493
views
maps from labelled configuration space to section space / iterated loop space
In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3:
for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle $T(M)$...
- 2,213
1
vote
1
answer
165
views
Group completion of labelled configuration space on Euclidean spaces
In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology
$$
\alpha_n: C(\mathbb{R}^n;X)\to \...
- 2,213
-2
votes
1
answer
289
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 ...
- 1,970
8
votes
1
answer
237
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 $U = \Omega^2 U$) ...
- 4,454
0
votes
1
answer
168
views
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
$$
\eta_n=\phi^{-...
- 1,970
19
votes
1
answer
2k
views
When does the free loop space fibration split?
This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that.
Let $X$ be ...
- 2,270
-2
votes
1
answer
279
views
stable splitting into a wedge sum [closed]
Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee _{k=1}^\infty Y_k.
$$
(1). Does this imply
$$
X\to \Sigma^tX\to \bigvee _{k=1}^\...
- 1,970
3
votes
1
answer
243
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\...
- 85
1
vote
0
answers
149
views
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 ...
- 627
3
votes
2
answers
436
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 ...
- 11.9k
5
votes
2
answers
566
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 ...
- 4,454
8
votes
1
answer
607
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}...
- 2,954
8
votes
1
answer
281
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 ...
- 1,771
16
votes
1
answer
905
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 ...
- 855
5
votes
1
answer
267
views
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 ...
- 163
7
votes
1
answer
679
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 ...
- 502
0
votes
1
answer
183
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 ...
- 1,970
1
vote
0
answers
177
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 $S^1\to X$ ...
- 523
3
votes
0
answers
333
views
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 ...
- 672
3
votes
1
answer
531
views
$E_{\infty}$ 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 $\mathcal{M}(j):=\Sigma_j$, ...
- 1,403
21
votes
1
answer
1k
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 ...
- 627