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Results tagged with homotopy-theory
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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.
29
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3
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The homotopy category is not complete nor cocomplete
I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What ar …
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28
votes
2
answers
2k
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Why study the p-completions of a space?
Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or Sulli …
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21
votes
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answer
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Maps out of Eilenberg-Mac Lane Spaces
Is anything known about the maps out of an Eilenberg Mac-lane Space $K(G,n)$?
Obviously I'm interested in extensions of Miller's resolution of the Sullivan conjecture, that $Map_*(K(G,1),X)\simeq\ast …
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15
votes
3
answers
462
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How stable is the top cell of a Lie group?
It is well known that the fundamental class of a compact Lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equ …
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14
votes
3
answers
630
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Strøm model structures on the category of simplicial sets
Let $X,Y$ be simplicial sets. A simplicial homotopy is a simplicial map of the form $h:X\times\Delta^1\rightarrow Y$. There are two distinguished maps
$$
in_0:X\cong X\times\Delta^0\xrightarrow{1\time …
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10
votes
1
answer
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Clutching functions and Classifying maps
Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C_+X\cup_X C_-X$. Then there are trivialisations of the restrictions $E|_{C_+X}\cong C_+X\times G$, $E|_{C_+ …
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9
votes
0
answers
320
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Samelson Products in $SO(n)$
Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most genera …
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8
votes
0
answers
253
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What is known about maps between loop spaces of Spheres? - Reference request
What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values o …
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8
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The James and Morse filtrations of homotopy groups
Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain …
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7
votes
3
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Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?
I believe that $S^1\vee S^1$ is the Eilenberg-Mac Lane space $K(\mathbb{Z}\ast\mathbb{Z},1)$. One can prove this by constructing its universal cover and observing that it is contractible.
My question …
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7
votes
1
answer
505
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Replacing the Fibre of a Fibration
This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature.
Let $p:E\rightarrow …
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6
votes
1
answer
292
views
Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$
Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\ri …
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5
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164
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Identifying a cohomology class arising from a Postnikov decomposition of BU(2)
For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface …
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4
votes
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231
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The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group
Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping …
- 5,596
4
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0
answers
87
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The order of $im(\nu'_*)\subseteq \pi_*S^3$
The 3-sphere $S^3$ has homotopy 2-exponent 4. That is, any 2-torsion element $\alpha\in\pi_*S^3$ has order at most 4. This bound is sharp, for example the Blakers-Massey element $\nu'\in\pi_6S^3$ has …
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