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.
159 questions
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Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
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Reference for computing cohomology of $Q_nS^0$ (the $n$-th component of infinite loop space)?
$Q_nS^0=\varinjlim_k(\Omega^kS^k)_n$, where $(\Omega^kS^k)_n$ refers to homotopy classes of maps $(S^k,\ast)\to (S^k,\ast)$ of degree $n$. I already know the case of $n=0$.
Is there any reference ...
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Loop space, parametrization equivalence and the issue of giving a topology
This question has been motivated by p.165 of this book.
As in the cited link above, we consider the following space of paraemtrized piecewise $C^1$ loops
\begin{equation}
X:= \Bigl\{ x : [0,1] \to \...
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String cobracket and co-Hochschild homology
Let $M$ be a closed oriented manifold and take a field of char. zero to be the ground ring. String Topology gives, to the homology $H_\bullet(LM)$ of the free loop space of $M$, the structure of ...
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Loop-space functor on cohomology
For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$.
More concretely, $\omega$ is given by the Puppe sequence
$$\...
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Homotopy equivalence between certain loop spaces
I've been reading some papers carefully, with their proofs (Notations are given at the end).
The following comes from "Braids, mapping class groups and categorical delooping" by Song & ...
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Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is ...
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Stable splitting of $\Omega SU(n)$
The space $\Omega SU(n)$ is homotopy-equivalent to $SL_n(\mathbb{C}[z,z^{-1}])/SL_n(\mathbb{C}[z])$. Using this, Steve Mitchell introduced a filtration of $\Omega SU(n)$ by subspaces $F_k$ which can ...
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Alternative construction for the loop space (?)
There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
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How now to study operads in homotopy theory?
There is a great introduction by May, "The Geometry of Iterated Loop Spaces". I really enjoy reading it, but it was written 50 years ago and contains outdated technical details related to ...
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The space of immersions of a loop in a surface
Let $\Sigma$ be a compact oriented surface with boundary and $L = \mathrm{Imm}(\bigsqcup_{i=1}^n S^1,\Sigma)$ the space of all generic (i.e. transversally and at most doubly intersecting) immersions ...
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Delooping a weak $E_1$ map by bar construction
Consider based maps $f : X \to Z$ and $g : Y \to Z$, which induces the following map at the based loop space level : $$\theta := \mu_Z \circ \big(\Omega f \times \Omega g) : \Omega(X\times Y) = \Omega ...
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Homotopy type / Homology of the free loop space of aspherical manifolds
Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
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Goresky-Hingston product on cohomology of the relative free loop space on $S^1$
I'm interested in the computations of the Goresky-Hingston product (defined https://arxiv.org/abs/0707.3486)
on the cohomology of the relative free loop space on the circle (or better yet, their ...
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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 ...
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Are these two concepts of a differential form on the loop space equivalent?
Notation:
Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$.
In the context of loop space homology and the supersymmetric path ...
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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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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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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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Bar constructions of $A_\infty$-algebras and rectifications
Let $\mathscr{C}_1$ be the little 1-cubes operad. If $X$ is an algebra over $\mathscr{C}_1$, I can think of (at least) two ways how to deloop it:
I can consider its two-sided bar construction $B_\...
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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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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 ...
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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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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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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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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 $X\to LX$. ...
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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 ...
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Eilenberg–Moore equivalences for $C_*(\Omega M)$
Let $M$ be a nice connected topological space (I'm actually interested in manifolds) with base point $p$ and let $\pi: E \to M$ be a fibration. Then chains on the fiber $F$ at $p$, $C_*(F)$, become a ...
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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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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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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 $X$...
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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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What are the obstacles for a complex to be a space of loops?
It is known that any space of loops is an H-space. So my question has two parts:
What are the obstacles for a complex to be an H-space? Is there any hope to somehow reasonably classify/characterize ...
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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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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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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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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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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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Simplicial realization of the circle action on the free loop space
Given a simply connected topological space $X$, it is well known that its free loop space $LX$ has cohomology being the Hochschild homology of the singular cochains [1]:
$$HH_\bullet(S^\star X) \simeq ...
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The free smooth path space on a manifold
Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map
$$
PM \to M \times M .
$$
Question ...
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Low-Dimensional Spaces with High-Dimensional Homology
Barratt-Milnor Spheres $X_n$ are spaces with finite topological dimension $n$ but which have non-vanishing singular homology in arbitrarily high dimensions. Here, they prove that if $n > 1$ then ...
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Reconciling the affine grassmannian and the based loop group
I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...
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Delooping the quotient space $SU/SU(n)$
Let $SU$ denote the infinite unitary group. Does the quotient space $SU/SU(n)$ admit a delooping $X$? One could also ask that this space $X$ sit in a fiber sequence $BSU(n)\to BSU\to X$, but this is ...
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The cohomology groups of $\Omega U(n)$
Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?
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$E_\infty$-space structure of $B\mathrm{GL}(\mathbb S_{(p)})$
In Geometric Topology - Localization, Periodicity, and Galois Symmetry by Dennis Sullivan, we can read that there is a decomposition
$$B\mathrm{SL}(\mathbb S_{(p)})\times K((\mathbf Z_{(p)})^\times)\...
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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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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^...
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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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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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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 ...
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