# Tagged Questions

802 views

### 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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### Proving that a space cannot be delooped.

Suppose we have some pointed connected topological space $X$. How can we determine if there exists a space $BX$, called delooping of $X$, such that its space of based loops $\Omega BX$ is homotopy ...
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### Are loop spaces of homotopically equivalent spaces homotopically equivalent? [closed]

Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces. Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence? Here, loop spaces ...
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### Free Loops, Moore Paths and the Borel Construction

My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homotopy equivalent ...
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### Loop space of a category

This seems like it should be a "standard" thing, and I think I remember even seeing it somewhere, but I can't remember where. Let $C$ be a small category. Is there a category $\Lambda C$ whose nerve ...
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### Homotopy type of the self-homotopy equivalences of a bouquet of spheres

Before I state the questions I have in mind, let me give some background. If one considers $S^2$ then it is known due to Kneser that $\textrm{Homeo}^{+}(S^2)$ has the homotopy type of $SO(3)$. By ...
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### Uniqueness of loop spaces

Suppose X is a loop space; by this we mean there is some space $Y$ with $\Omega Y \simeq X$. Under what assumptions is (the homotopy type of) $Y$ unique? As has been pointed out below, the ...
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### Classifying spaces of E_1 - spaces

Hello, I try to understand aspects of homotopy coherence, in particular "recognition principle" of May. About the following I did not think a lot, but I decided to ask here anyway, so to save ...
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### What are the algebras over $\Omega^k\Sigma^k$ ?

Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair $$\Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k,$$ where $\Sigma^k$ is the $k$-th supension functor and ...
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### Proof of the ''trangression theorem''

Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
Let $f: X \to Y$ be a fibration of pointed Kan complexes, and let $F$ be the fiber. Question: How do you prove that the following diagram of homotopy groups commutes?: \$\pi_n(Y) \to ...