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.
159 questions
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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 $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ ...
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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 ...
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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
...
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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-...
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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 ...
31
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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 ...
29
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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 ...
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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 ...
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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 ...