# Tagged Questions

(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.

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### Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
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### Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible. The problem is from the following material. It contends that the result is by standard Morse theory....
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### Iterating the suspension-loop adjunction in two different ways

Let $X$ be a sufficiently nice topological space (i.e. an object of a category of spaces where the reduced suspension-loops, $(\Sigma, \Omega)$, holds.) There are two directed systems of spaces one ...
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### Clarification regarding little n-discs operads

I am reading the wiki page on operad theory and I am trying to figure out how exactly those "Little something" operads work which are mentioned there. Specifically, I am having a hard time, despite ...
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### configuration-spaces and iterated loop-spaces

In the paper Configuration-Spaces and Iterated Loop-Spaces. Graeme Segal, page 213-214, it is obtained that the labelled configuration space $C_n$ is homotopy equivalent to a topological monoid $C'_n$....
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### homomorphism of $H$-spaces between a monoid and loop space of its classifying space

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. The geometric realization $BM=|N_\bullet(M)|$...
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### pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...
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### question about “Homology fibrations and the group completion theorem”

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In ...
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### canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Timelike Loop Spaces as Projective Null Twistor Spaces

I have already asked this question in physics.stackexchange, but have not got any answers, so I have decided to ask it here. Let $\mathcal{M}$ be a spacetime, and let $\Omega\mathcal{M}$ denote the ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $f \text{ and } g$...
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### 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. http://topospaces.subwiki.org/wiki/...
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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. ...
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### 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 ...
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### 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 ...
### 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 ...
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 ...