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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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
$$\...
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Homotopy equivalence between certain loop spaces
I've been reading some papers carefully, with their proofs (Notations are given at the end).
The following comes from "Braids, mapping class groups and categorical delooping" by Song & ...
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Alternative construction for the loop space (?)
There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
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The space of immersions of a loop in a surface
Let $\Sigma$ be a compact oriented surface with boundary and $L = \mathrm{Imm}(\bigsqcup_{i=1}^n S^1,\Sigma)$ the space of all generic (i.e. transversally and at most doubly intersecting) immersions ...
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Homotopy type / Homology of the free loop space of aspherical manifolds
Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
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Delooping a weak $E_1$ map by bar construction
Consider based maps $f : X \to Z$ and $g : Y \to Z$, which induces the following map at the based loop space level : $$\theta := \mu_Z \circ \big(\Omega f \times \Omega g) : \Omega(X\times Y) = \Omega ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$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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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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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 ...
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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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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 ...
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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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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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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 ...
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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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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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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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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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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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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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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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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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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 ...
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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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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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