(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.

learn more… | top users | synonyms

0
votes
0answers
13 views

On loop space of a product

[consider good spaces so the objects and question make sense] Who was the first to prove that the loop space of a product is homeomorphic to the product of the loop spaces? $$\Omega(X\times Y)\approx ...
3
votes
1answer
64 views

For compact surface $M$ and loop $f$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such that $f \notin \ker(\phi)$

Why is this sentence true? For every not nullhomologous loop $f$ without selfintersections on orientable compact surface $M$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such ...
0
votes
0answers
34 views

existance of loop with finitely many point of intersection

for every loop on compact orientable surface exists freely homotopic loop with finitely many points of intersection. I see that it have to be true, but I can't prove it. I know Thom's theorem, Sard's ...
0
votes
0answers
38 views

lifting a closed curve

Is it always true (because of covering spaces has homotopy lifting property)? loop $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve. or we have to ...
3
votes
3answers
170 views

How to explain why free loop space, or based loop space, is infinite dimensional to non-math people.

I am giving a math talk to non-mathematicians. I was wondering how to explain how the free loop space, or based loop space, of a topological space is infinite dimensional so that a non-mathematician ...
3
votes
1answer
57 views

Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
2
votes
0answers
91 views

Conditions for a Topological space to be a Spectrum

I'm looking for conditions for a topological space $X$ to be a Spectrum. A topological space $X$ is a spectrum if it can be delooped infinitely (more accurately, «double-infinitely»). Some ...
2
votes
0answers
39 views

Clarification about a double delooped H-space.

I've just started reading J.P. May's book The Geometry of Iterated Loop Spaces and am misunderstanding something. Somewhere, it's asserted that if an H-space X can be delooped twice, the its ...
4
votes
2answers
89 views

Derivation and meaning of a long exact sequence of a Homotopy Groups for pairs of spaces

I read that there are long exact sequences of homotopy groups for each pair of pointed spaces $(X,A,x_{0})$. Now I know that for an exact sequence that, as the example below denotes $f \text{ and } ...
3
votes
0answers
275 views

Action of the Fundamental Group on Higher Homotopy Groups.

First: here are a couple links of which I am looking at. I try to add the relevant information (at least to my understanding) from them. ...
2
votes
0answers
60 views

Γ-spaces and operads

I'm looking for a comprehensible reference that explains how $\Gamma$-spaces are related to $E_{\infty}$-operads. I've found some old publications but was hoping there are better references out there. ...
1
vote
1answer
69 views

What kind of information is given by the n-th homotopy group for n>=2?

How i see this the homotopy group of order 1 is giving information about the "holes" in a topological space. In that way what kind of information is giving the homotopy group of order 2 or n in ...
5
votes
1answer
206 views

Why is the recognition principle important?

The recognition principle basically states that (under some conditions) a topological space $X$ has the weak homotopy type of some $\Omega^k Y$ iff it is an $E_k$-algebra (ie. an algebra over the ...
5
votes
3answers
618 views

Showing that the loopspace $\Omega S^{\infty}$ is homotopic to $S^{\infty}$.

Showing that the infinite dimensional sphere $S^{\infty}$ is contractible is rather easy by constructing an explicit contraction (Hatcher gives a nice one). I thought it might be a nice exercise to ...
3
votes
1answer
180 views

Non-vanishing of homology of loop spaces

One of the answers to this MO question implies that loop spaces of $S^n$ for $n>1$ have non-zero homology in arbitrarily high degree. Is there any simple (or, better yet, geometric) way to prove ...
4
votes
1answer
230 views

Associative up to homotopy

I'm reading Adams' book Infinite Loop Spaces. He explains that the product map on a loop space isn't associative, but it is associative up to coherent homotopy. I'm confused about the coherent part of ...
4
votes
2answers
549 views

Homology of the loop space

Let $X$ be a nice space (manifold, CW-complex, what you prefer). I was wondering if there is a computable relation between the homology of $\Omega X$, the loop space of $X$, and the homology of $X$. I ...
8
votes
1answer
674 views

What's the loop space of a circle?

Is it true that the loop space of a circle is contractible? Consider the long exact sequence in homotopy for the path fibration $\Omega S^1 \rightarrow \ast \rightarrow S^1$ shows all homotopy groups ...
6
votes
1answer
312 views

Does a map of spaces inducing an isomorphism on homology induce an isomorphism between the homologies of the loop spaces?

That is, let $f:X \rightarrow Y$ be a map of spaces such that $f_*: H_*(X) \rightarrow H_*(Y)$ induces an isomorphism on homology. We get an induced map $\tilde{f}: \Omega X \rightarrow \Omega Y$, ...