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Questions tagged [loop-spaces]

The loop space $Ω_X$ of a pointed topological space $X$ is the space of based maps from the circle $\mathbb S^1$ to $X$ with the compact-open topology.

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2 votes
3 answers
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parameterizing polynomial loops in $\mathbb{C}^\times$

Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L_{poly}\mathbb{C}^\times$ of loops that look like $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ ...
solbap's user avatar
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17 votes
2 answers
1k views

What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
skupers's user avatar
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14 votes
2 answers
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Explicit cocycle for the central extension of the algebraic loop group G(C((t)))

Let $G$ be a simple Lie group and let $G(\mathbb{C}((t)))$ be its loop group. The Lie algebra $\mathfrak{g}[[t]][t^{-1}]$ has a well known central extension (see e.g. Wikipedia) given by the cocycle ...
André Henriques's user avatar
12 votes
1 answer
560 views

Are centrally extended p-adic groups defined over F_1?

Let G be a semisimple algebraic group. Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor G : Rings → Groups by the second algebraic K-...
André Henriques's user avatar
3 votes
0 answers
267 views

Maps of loop spaces with infinity-bounded differential.

I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally: In the following piece-wise smooth means smooth on ...
Thomas Kragh's user avatar
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31 votes
3 answers
4k views

Algebras over the little disks operad

Hello, The so-called "recognition principle" of Boardman-Vogt and May leaves me unsatisfied. My problem is the following: The "recognition principle" says that every "group-like" algebra over the ...
Oblomov's user avatar
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29 votes
5 answers
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Topologists loops versus algebraists loops

Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also ...
Peter McNamara's user avatar
26 votes
2 answers
2k views

Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?

There are two ways to define smooth mapping spaces and I want to know how they compare. Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
Chris Schommer-Pries's user avatar
20 votes
5 answers
3k views

Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
Reid Barton's user avatar
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