Skip to main content

Questions tagged [loop-spaces]

(Stub) The loop space is the function space consisting of all continuous maps from the circle into a topological space; the function space is equipped with the compact-open topology. It is studied in topology, especially homotopy theory.

Filter by
Sorted by
Tagged with
1 vote
0 answers
52 views

Loop spaces and smash product

For context, my question stems from a claim of Waldhausen (page 342) where from a map $$|wS.\mathcal{A}|\wedge |wS.\mathcal{B}|\to |wwS.S.\mathcal{C}|.$$ He claims it induces a map $$\Omega|wS.\...
DevVorb's user avatar
  • 1,433
3 votes
0 answers
77 views

Find $\mathscr X$ such that $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $\mathscr X(\Omega G)$ for any $G$

For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of ...
მამუკა ჯიბლაძე's user avatar
5 votes
1 answer
76 views

Is the Moore Loop space of a well-pointed space well-pointed?

Suppose that $X$ is a well-pointed topological space (i.e. the basepoint is cofibered, not necessarily closed). Is the Moore loop space/space of measured looks $\Omega_MX$ well-pointed? I have seen ...
Thorgott's user avatar
  • 12.4k
5 votes
0 answers
93 views

Why is an (algebraic) loop group called a "loop group"?

Let $G$ be an algebraic group over a field $k$. Then we can define the loop group $LG$ to be the sheaf which takes a $k$-algebra $R$ and spits out $G(R((t)))$. My question is, why is this called the ...
Calculus101's user avatar
0 votes
0 answers
36 views

Definition of differential forms on loop space and the space of figure eights

One can define differential forms $\Omega^*(LM)$ on the (smooth) free loop space $LM$ by giving $LM$ the structure of a diffeological space or a Chen differential space. A diffeological space $X$ is ...
blue's user avatar
  • 755
5 votes
2 answers
707 views

cohomology of free loop space on $S^3$

Let $LS^3=\text{maps}(S^1,S^3)$ denote the free loop space on $S^3$. I want to compute the cohomology of this space. I think that it should be $\mathbb{Z}$ in all degrees except degree one where it ...
Womm's user avatar
  • 496
1 vote
1 answer
61 views

The fundamental group of the loop space of $(X,x_0)$ with the base point chosen not to be the constant loop in $x_0$

My question is pretty simple although I have not been able to find an answer yet: Let $c_{x_0}\in\Omega(X,x_0)$ denote the constant loop in $x_0$. Then, by standard homotopy theoretical arguments, we ...
Mathematics enthusiast's user avatar
1 vote
1 answer
133 views

What are the loops in BG?

I'm having trouble with the following elementary(?) thing, but its confusing me alot. For a category $\mathcal{C}$, we can define the classifying space in terms of nerve. Similarly, for a topological ...
May's user avatar
  • 255
1 vote
0 answers
59 views

A couple of questions on homotopy fibrations. [closed]

I have several questions, but they all are mostly definition-centered and, I assume, are easy for a person with good understanding of fibrations (unfortunately, I am not one). Since there are several ...
Haldot's user avatar
  • 790
2 votes
0 answers
87 views

Reference for Stasheff's Associahedra Operad

I am currently reading up on operads and am more than confused about the Stasheff operad. It is completely unplausible that I am the first student, who feels this way, so I hope to find a reference ...
Jonas Linssen's user avatar
1 vote
0 answers
94 views

Examples of Hilbert manifolds?

I came across the definition of Hilbert manifold on Wikipedia (https://en.wikipedia.org/wiki/Hilbert_manifold) and was hoping to collect some (infinite-dimensional) examples. I'd be particularly ...
Brendan Mallery's user avatar
0 votes
1 answer
83 views

Explicit examples of loop spaces

I'm reading Spanier, and I have understood that loop spaces are examples of H - groups. However, I'm unable to explicitly work out an example of a non-trivial loop space. My question: Give an example ...
Sundara Narasimhan's user avatar
0 votes
1 answer
117 views

about loop and suspension

I am trying to understand the following construction. $X$ is a pointed CW complex. Define $Q(X) := hocolim_{n} \Omega^n \Sigma^n (X)$. Using the loop-suspension adjunction, we get maps like $X \to \...
Algtop's user avatar
  • 25
0 votes
1 answer
383 views

Explicit description of classifying space as a "delooping"?

Consider the "classifying-space functor" $\mathscr{B} : \mathrm{TopGrp} \rightarrow \mathrm{Top}$, constructed in the standard way as a geometric realization of a nerve (in TopCat if ...
I.A.S. Tambe's user avatar
  • 2,441
5 votes
0 answers
129 views

A map from the symmetric algebra generated by the first cohomology to the cohomology groups.

Let $X$ be a topological space (a CW complex) and $\mathbb{P}^d$ here represents the complex projective space. Consider the space of base-point preserving maps from $X$ to $\text{Sym}^n(\mathbb{P}^d)$,...
user127776's user avatar
  • 1,364
1 vote
0 answers
80 views

Why is the induced map on cofibers in the loop space homotopy pullback the counit?

Let $X$ be a space and consider the homotopy pullback diagram $$\require{AMScd} \begin{CD} \Omega X @>>> *\\ @VVV @VVV \\ * @>>> X \end{CD}$$ Taking the induced map on cofibers, we ...
merle's user avatar
  • 585
0 votes
0 answers
55 views

Accessible Citation for the associative based loop space

John Moore defined a based loop space using maps of intervals [0,r]. It was defined in a samizdat version of the Milnor-Moore paper on Hopf algebra, but was cut in the published version. Is there a ...
Jim Stasheff's user avatar
2 votes
1 answer
92 views

Is there an analogue of the loop space for homology? [closed]

Is there an endofunctor $U: \mathrm{Top} \to \mathrm{Top}$ (or from some good subcategory) such that $H_n(UX) = H_{n+1}(X)$ for any $n \geq 1$
Arshak Aivazian's user avatar
1 vote
0 answers
68 views

Explicit description of $\pi_1(M)$-action on the relative homotopy groups $\pi_k(\Omega_0M,M)$

Let $M$ be a path-connected space (say, a connected manifold) and denote by $\Omega_0 M$ the based loop space component of the constant loop $x_0$. If $c \colon M \to \Omega_0 M$ denotes the obvious ...
noctusraid's user avatar
  • 1,666
3 votes
1 answer
113 views

Why is the fundamental group not a topological group?

It is widely known that group operations in $\Omega X$ are generally not consistent with the natural topology (see, for example, https://arxiv.org/pdf/1105.6363.pdf). Moreover, if the operations of ...
Arshak Aivazian's user avatar
2 votes
1 answer
164 views

Inclusion of space loop is a Serre fibration

Let $i: X \longmapsto Y$ be an inclusion, and define $$\Omega_X Y:= \left\lbrace \gamma : I \longmapsto Y : \gamma(1) \in X \right\rbrace$$ Is it true that the map $$\bar{i} : \Omega_x Y \longmapsto Y$...
jacopoburelli's user avatar
2 votes
0 answers
69 views

Are loop space bundles fibrewise equivalent to principal bundles?

Suppose given a map $f:X\to Y$ with homotopy fibre a loop space $\Omega Z$. Then there is a homotopy equivalence $\Omega Z\simeq G$ for some topological group $G$. Does there exist a principal $G$-...
მამუკა ჯიბლაძე's user avatar
8 votes
1 answer
135 views

Is the space of complex structures a loop space?

Define the space of (normalized) complex structures $\mathcal{J}_{2k}$ on $\mathbb{R}^{2k}$ as the orthogonal transformations in $SO(2k)$ that square to minus the identity. My question is if there ...
SourcedDirect's user avatar
4 votes
2 answers
435 views

Loop-space uniquely characterized by shifting of homotopy groups?

Let $(X,x_{0})$ be a topological space and let $(Y,y_{0})$ be another topological space such that for homotopy groups $\pi_{i}(Y,y_{0}) \cong \pi_{i+1}(X,x_{0})$ for all $i \geq 0$. Is it then true ...
Sunny Sood's user avatar
1 vote
0 answers
69 views

Homotopy in the universal covering of a connected component of the loop space

Let $M$ be a compact manifold and $L_0,L_1$ two compact submanifolds. Let's definte $P(L_0,L_1):=\{\gamma :[0,1]\rightarrow M|$smooth $, \gamma(0)\in L_0, \gamma(1)\in L(1)\}$. Now fix $\gamma'\in P(...
Someone's user avatar
  • 4,747
0 votes
1 answer
62 views

Fourier expansion of (small perturbations of constant) loops

In this paper on page 4 the authors write: We would like to apply such a construction to the tangent bundle of a free loopspace. [...]. However, such a splitting does exist in a neighborhood of the ...
Lynn Otto's user avatar
  • 376
1 vote
0 answers
87 views

Reference for first appearance of Samelson product

Samelson noticed that the commutator in any based loop space induced the Pontrjagin product on the homology of that loop space. What is the original reference where the name Samelson product is ...
Jim Stasheff's user avatar
3 votes
1 answer
64 views

How can I show that $X_l$ and $X_{f \circ l}$are homotopy equivalent?

Let $(X, \mathcal{T}_X)$ be a topological space. Let $l$ be a loop so that $p \in X$ is the base point and $f: X \to X$ a continuous function, where $f \simeq id_X$ ($f$ and the identity function is ...
Chris's user avatar
  • 359
4 votes
2 answers
129 views

Is the action functional "locally well defined up to an additive constant"?

I'm looking at an arbitrary symplectic manifold $(P, \omega)$, and the action functional on the set of contractible loops $\Lambda P:$ $$A: \Lambda P \to \mathbb{R}, \hspace{5pt} A(z) = \int_{D^2} \...
Matija Sreckovic's user avatar
3 votes
0 answers
140 views

Classifying spaces of $E_1$-spaces

I'm a newbie trying to understand May's recognition principle on $E_1$-spaces. In May's paper The Geometry of Iterated Loop Spaces, the classifying space of an $E_1$-space $X$ is defined to be $B(\...
psy's user avatar
  • 31
2 votes
0 answers
68 views

How to reconstruct a closed loop like a circle or a loop with continuous curvature change i.e. arbitrary shape by using the Frenet-Serret equation

I want to reconstruct a closed loop in 2D or 3D space (depending on whether there is torsion or not) through calculating the Frenet-Serret equations the Frenet-Serret equations, just like the figure ...
lindar's user avatar
  • 21
3 votes
0 answers
363 views

Mahowald-Hopkins theorem

I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following: Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom ...
elidiot's user avatar
  • 2,159
0 votes
0 answers
254 views

Why are loop spaces $\Omega Y$ examples of grouplike spaces?

Here is my definition for a grouplike space or an $H-$group from "Introduction to homotopy theory" by Martin Arkowitz: And here is a question I found here: Every loop space $(\Omega Y,w_0)$ ...
user avatar
0 votes
1 answer
49 views

How does multiplication in $\Omega X$ induce multiplication for chains of $\Omega X$?

I have a CW complex $K$ and a loop space $\Omega K$. By $Q(\Omega K)$ I denote the group generated by the singular cubes of $\Omega K$. Then the book says that "the multiplication in $\Omega K$ ...
Haldot's user avatar
  • 790
5 votes
0 answers
164 views

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

I am studying J. Strom's Modern Classical Homotopy Theory. In chapter 4 he proposes the following exercise Let $ X $ be a path-connected non-contractible space. (1) It is possible that $ X \simeq \...
CNS709's user avatar
  • 1,667
1 vote
1 answer
197 views

Connected components of free loop space

Let $X$ be a topological space. And let $\Lambda X=\mathrm{Top}\left[S^1,X\right]$ be the space of continuous loops in $X$. Then how do we calculate $\Pi_0\Lambda X$, the set of connected components ...
Chetan Vuppulury's user avatar
3 votes
1 answer
60 views

On how to compute $H_*(\Omega^2 S^{n+2}, \mathbb Z/2)$ and its Steenrod operations

I've been trying to understand the calculation of $H_*(\Omega^2 S^{n+2},\mathbb Z/2)$ as an algebra, together with the dual Steenrod operations on it which lower the degrees. I've spent many hours ...
elidiot's user avatar
  • 2,159
1 vote
0 answers
38 views

Galois Cohomology and Loop Groups

I am trying to understand problem 8.5 in Kac's Infinite dimensional Lie algebras. It goes as follows. Let $G$ be a semisimple algebraic group, let $\alpha$ be an automorphism of $G$ of order $m$, and ...
Marc Besson's user avatar
0 votes
0 answers
83 views

Why is the attaching map for the top cell in the connected sum denoted by "sum" of attaching maps of top cells in the manifolds?

In a book I read: $M$ and $N$ are $d$-dimensional manifolds. Let $\widetilde{M}$ be the $(d−1)$-skeleton of $M$, or equivalently, $\widetilde{M}$ is obtained from $M$ by removing a disc in the ...
Haldot's user avatar
  • 790
0 votes
0 answers
73 views

What is meant by "adjoint" map to loop space?

I have continuous map $f:S^2\to X$. Then there is some kind of an "adjoint" map $\hat{f}:S^1\to\Omega X$. What is it?
Haldot's user avatar
  • 790
1 vote
1 answer
259 views

Proof that every path in U is homotopic

Let $U$ a connected space and given $x,y,x',y'$ four points $\in U $ and it says that every statement implies the other : 1) Every path ( in $U$ ) from $x \longrightarrow y $ is homotopic in $...
user avatar
3 votes
1 answer
347 views

Grassmannians and $\mathrm{GL}(n,\mathbb{R})$

Let $\mathrm{Gr}_n$ denote the infinite real Grassmannian of $n$-planes in $\mathbb{R}^\infty$. This is a classifying space for real vector bundles, in the sense that (for paracompact $B$) $$ [B, \...
SvanN's user avatar
  • 2,317
2 votes
1 answer
685 views

What is the fiber of the path space fibration?

Schematically, I understand the path space fibration $PX$ over some path-connected, pointed topological space $X$ with base point $x_o$ as: $$\Omega X \hookrightarrow PX \twoheadrightarrow X,$$ where ...
k-t's user avatar
  • 23
2 votes
1 answer
128 views

$\langle X,\Omega^2 K\rangle$ is abelian

How can I show that $\langle X,\Omega^2 K\rangle$ is abelian? Here $X$ and $K$ are topological spaces with basepoints $x_0$ and $k_0$, respectively. The loopspace $\Omega^2 K$ is the set of all maps $...
blancket's user avatar
  • 1,850
1 vote
0 answers
72 views

Under what conditions can one deloop the free loop fibration?

Inspired by the MO question Homotopy extension of $E_\infty$-spaces. Sending a map $f:S^1\to X$ to $f(\text{basepoint})$ gives a fibration $\Lambda X\to X$ with fiber $\Omega X$, where $\Lambda X$ is ...
მამუკა ჯიბლაძე's user avatar
0 votes
1 answer
85 views

Are the base points of freely homotopic loops guaranteed to be in the same path-component?

We have two loops $f,g$ in a space $X$ with $f(0)=f(1)=x$ and $g(0)=g(1)=y$ (with $x$ not necessarily equal to $y$), which are homotopic, that is there exists a "path" made of loops (or are any paths ...
B.Swan's user avatar
  • 2,469
0 votes
0 answers
170 views

Definition of an infinite loop space

Let $E_0$ be a topological space. Are the following two definitions of an infinite loop space $E_0$ equivalent? Definition 1: For each $n>0$, there exists a topological space $E_n$ such that $E_{n-...
Yuhang Chen's user avatar
4 votes
1 answer
348 views

Is the homology of the based loop space of a compact globally symmetric space a polynomial ring?

Let $X$ be a space. Then the homology group $H_*(\Omega X;\mathbb{Q})$ of the based loop space of $X$ is a $\mathbb{Q}$-algebra with the Pontryagin product given by loop concatenation. When $X=G$ is ...
ChiHong Chow's user avatar
0 votes
1 answer
21 views

Let x,y,z,u be loops in a point p: x•y and z•u are homotopic and y is homotopic to u then x is homotopic to z

I have the question above and now I am trying to prove that x is homotopic to z. My Idea is as follows: Let $H(x,t)$ be the homotopy between x•y and z•u because y and u are homotopic I can ...
Jamal's user avatar
  • 35
3 votes
0 answers
91 views

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 ...
Mattia Coloma's user avatar