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Results tagged with homotopy-theory
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user 360
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.
3
votes
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Geometric Realizations of Simplicial Based Spaces
This is going to be the same example the one from this previous question: https://mathoverflow.net/a/171423/360
Let $A = \Bbb N$ and $B = \{0,1,1/2,1/3,1/4,\dots\}$, with the map $f: X \to Y$ given a …
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7
votes
Accepted
Do homotopy pullbacks commute with homotopy orbits (in spaces)?
Yes. A sketch:
Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibratio …
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25
votes
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Eilenberg-Mac lane spaces and a generalization
Assuming $m > n$, there is a method for classifying such spaces using a technique from the Postnikov tower. Namely, such a space has a map $X \to K(G,n)$ inducing an isomorphism on $\pi_n$, and if we …
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5
votes
$\pi_0$ of a cosimplicial space
This does not have to be surjective. In all cases, $\varprojlim \pi_0(X^n)$ is the same as the equalizer of the two maps $\pi_0 X^0 \rightrightarrows \pi_0 X^1$ (the rest of the diagram is redundant f …
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13
votes
Accepted
Do topological spaces form a full subcategory of spectra?
As noted in the comments, this is most definitely false in homotopy categories: the set $[S^1, S^0]$ of homotopy classes of based maps is trivial, but the set $[\Sigma^\infty S^1, \Sigma^\infty S^0]$ …
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8
votes
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Homology of localisations of spectra
I'm going to phrase in terms of $H$-homology instead of cohomology.
If $H$ is a finite spectrum, smashing with it always commutes across the limit. More generally, if you express $H$ as a (homotopy …
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7
votes
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$\mathbb{Z}/2$-action on spectra given by inversion
There's often no self-map of $S$ which induces multiplication by $(-1)$, because it's usually not cofibrant-fibrant. There's no homotopical obstruction to it existing, because the map $E \mapsto E \w …
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10
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Killing the torsion in homotopy
No, there is no such procedure. The problem is that attaching new cells can change the nontorsion in higher homotopy degrees and make it more divisible than it used to be.
One example: If X is BSp, …
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12
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What would be the ramifications of homotopy theory being as easy as homology theory?
Homology groups and homotopy groups are two sides of the same story. Homotopy groups tell us all the ways we can have a map Sn → X, and in particular describe all ways we can attach a new cell to our …
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5
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Good reference for homology of $K(\mathbb{Z}, 2n)$?
For a reference, you might see this paper of Birgit Richter. A rough outline follows:
Since $X = K(Z,n)$ can be made a commutative topological monoid per Ben Wieland's answer, its singular chain com …
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14
votes
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Construct a CW complex with prescribed homotopy groups and actions of $\pi_1$.
One method is as follows.
Construct $Y_i = K(\pi_i, i)$ for $i \geq 2$ as a based space with an action of the group $\pi_1$. You can do this manually (by attaching free orbits of cells along the gr …
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3
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proving that an inclusion map from a subcomplex is a homotopy equivalence
You can then apply this to id: (|X|,|A|) -> (|X|,|A|) to get a homotopy, rel A, from the identity of X to a retraction r: X -> A. This shows that A is a strong deformation retract of X and is in part …
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5
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Homotopy type of stabilizers
The short: no. E.g. let X be a topological group and G be the underlying discrete group of X, acting on X by left translation.
One standard hypothesis is the existence of slices. For some (hence an …
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3
votes
Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?
Stefan Schwede's "An unititled book project about symmetric spectra" covers, in chapter III, the projective levelwise and stable model structures on symmetric spectra in quite some detail (along with …
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15
votes
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Eckmann-Hilton for $A_{\infty}$-spaces?
EDIT: Here is a counterexample to the stated question.
I'm going to start with a topological group $G$ which is a product of Eilenberg-Mac Lane spaces. Specifically, I'm going to choose $G \simeq K( …
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