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Results tagged with homotopy-theory
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user 3995
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
10
votes
1
answer
652
views
BU with tensor product H-space structure
Hi,
I came across the space $BU_\otimes$ when struggling with twisted K-theory. Segal proved that this is an H-space, right? I have read a dozen times by now that the group $[X, BU_\otimes]$ consist …
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5
votes
1
answer
585
views
Delooping maps between H-spaces
Hi,
this question is related to my question here. Suppose, I have a topological group $G$ and an $A_{\infty}$-space $H$, which is a CW-complex. Furthermore, I have a map $\varphi \colon G \to H$, tha …
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8
votes
1
answer
682
views
Eckmann-Hilton for $A_{\infty}$-spaces?
Suppose I have a grouplike $A_{\infty}$-space $G$ that carries an additional structure as a topological group (which does not coincide with the $H$-space structure of the $A_{\infty}$-part). Denote th …
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3
votes
1
answer
371
views
Homotopy Units in $A_\infty$-spaces
Suppose I have an $A_{\infty}$-space $X$, such that its unit is only a unit up to homotopy. When the space is well-behaved (well-pointed? What is the weakest condition possible?), I can replace it wit …
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5
votes
2
answers
508
views
Weak operad and deloopings
Let $E$ be an operad in topological spaces. $E$ is usually called an $E_{\infty}$-operad, if all the spaces $E_n$ are contracticle. If $E$ acts on a space $X$, then by the recognition principle, $X$ t …
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3
votes
2
answers
272
views
classifying space of linear embeddings
Take the following small model for the category of finite-dimensional vector spaces and isomorphisms: The set of objects is $\mathbb{N}$ and the set of morphisms $Mor(n,m)$ is empty, if $n \neq m$ and …
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5
votes
1
answer
548
views
Classifying spaces of topological categories
I try to understand and compare the following facts about maps into classifying spaces of topological categories. Consider first the following definition cited from Moerdijk's Classifying Spaces and C …
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5
votes
1
answer
432
views
units in non-commutative ring spectra
Let $R$ be a connective (symmetric) ring spectrum. Let $GL_1(R)$ be the space of units of $R$, i.e. the union of the components of $\Omega^{\infty}(R)$ corresponding to the units of $\pi_0(R)$. $GL_1( …
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6
votes
1
answer
419
views
Properties of coefficients of ring spectra
This is an awkwardly backwards question, but bear with me here: Suppose I have a graded ring $R$ with unit, which has an invertible element $u$ in degree $2$. The multiplicative formal group law $f(x …
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6
votes
2
answers
464
views
Two commuting operad actions
If the following is true, it is probably well-known to the experts. Nevertheless, I could not find a reference for it.
Suppose $P$ and $Q$ are $A_{\infty}$-operads in topological spaces and $X$ is a …
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3
votes
2
answers
464
views
Nonabelian cohomology via crossed modules
Let $G$, $H$ be topological groups and let $t \colon G \to H$ be a homomorphism, such that $t \colon G \to H$ is a topological crossed module. For a topological space $X$ we can define the nonabelian …
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12
votes
2
answers
998
views
Twists of K-theory and tmf
I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How …
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9
votes
0
answers
461
views
Two constructions for BU×Z
Consider the following two ways of getting the zeroth space in the $K$-theory spectrum $BU \times \mathbb{Z}$:
1) Take the groupoid of finite dimensional complex inner product spaces with isometries …
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2
votes
0
answers
128
views
spaces of projections
Let $\mathbb{K}$ be the compact operators on a separable infinite dimensional Hilbert space. Denote by $\mathcal{P}(\mathbb{K})$ the space of projections in $\mathbb{K}$. If I am not terribly wrong he …
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4
votes
0
answers
328
views
good covers and simplicial maps
Let $X$ be a paracompact topological space and choose a good cover $U_i$ of $X$. Remember that a good cover is one that consists of open subsets, such that each set $U_i$ is contractible and all finit …
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