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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.

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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 know that, almost by definition, the homotopy…
Thomas Rot
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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 of the loop space to be zero, and then…
Alex
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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 operad of the little $k$-cubes). This principle is…
Najib Idrissi
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Loop space of wedge sum of spheres

What is an explicit formula for $\Omega (S^n \vee S^m)$? I know that it follows from Hilton-Milnor theorem. But I don't quite understand it's formulation.
RedSea
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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 exists a space $X$ such that $\Omega (X) \simeq…
SourcedDirect
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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 $\tilde{f}: \Omega X \rightarrow \Omega Y$, where $\Omega X$ is the loop space of $x$. Does…
Tony
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Homology of free loop space

By rational homotopy theory, $H(\Lambda M; \mathbb{Q})$ is infinite-dimensional over $\mathbb{Q}$ if $M$ is simply-connected. Are there (non-simply-connected) examples when $H(\Lambda M; \mathbb{Q})$ is finite-dimensional? I am most interested when…
user39598
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Every loop space $(\Omega Y,w_0)$ has the structure of an $H$-group.

The most important example of an $H$-group is the loop space $(\Omega Y,w_0)$ of any pointed space $(Y,y_0)$. Let $\mu:\Omega Y\times \Omega Y\to \Omega Y; \;\; \mu(\alpha,\beta)=\alpha \star\beta$, where $\alpha \star\beta$ is the product of two…
M.A.
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Loop spaces have the homotopy type of a topological groups

Every based loop space has the homotopy type of a topological group. I would like to understand this fact, and this is what this question is about : why is it true, and how does one prove it? I think i have a proof of this fact (which I'll post…
Olivier Bégassat
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multiplication in H-space and loopspace of the H-space

Let $X$ be an $H-space$, and let a multiplication $,\cdot,$ be given, associative up to homotopy. Let $\Omega X$ be the loopspace of $X$ based at the identity and let the multiplication $ \circ $ on the loopspace be given by concatenating loops. We…
Hari Rau-Murthy
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Textbook on infinite loop spaces

I'm looking for a good update reference covering the material in first three chapters of "Adams, Infinite loop spaces" (specially construction of delooping functors and group completion) with exact definitions and complete proofs. Thanks!
Mostafa - Free Palestine
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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 this though. In his explanation he uses a…
Joe
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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 this claimed in multiple places, but have not been…
Thorgott
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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 loop group? If one takes $k = \mathbb{C}$, then is…
Calculus101
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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 should be zero. I have computed this using the…
Womm
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