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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.
17
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3
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Example of an unstable map between finite complexes which is the identity on homotopy but no...
Stably, phantom maps (nonzero maps which are zero on homotopy) exist, but it's not known if they exist between finite complexes (Freyd's Generating Hypothesis). Unstably, it's easy to find maps which …
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5
votes
1
answer
166
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Do simplicial join and product form a duoidal category structure?
The join $\ast$ and the product $\times$ are both important monoidal structures on simplicial sets, but the way they interact is not so simple. For instance, neither distributes over the other. Howeve …
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6
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244
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Is the natural geometric realization of symmetric simplicial sets homotopically correct?
Recall that the category of $\sigma Set$ of symmetric simplicial sets is the category of presheaves on $\Sigma$, the category of finite nonempy sets and all functions. The inclusion $v: \Delta \to \Si …
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5
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1
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312
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What can be said about the map $K(n)_\ast(X) \to K(n)_\ast(\tau_{\leq n}X)$ when $X$ is a fi...
Ravenel and Wilson showed that $K(\mathbb Z / p^j,q)$ is $K(n)$-acyclic for any $q \geq n+1$, and that $K(\mathbb Z, q)$ is $K(n)$-acyclic for $q \geq n+2$. It follows that $K(A,q)$ is $K(n)$-acyclic …
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5
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To what extent is homological localization determined by its values on $K(G,n)$'s?
Consider a homological localization $L$ of $p$-local spaces or (I think equivalently) a localization of $p$-local spectra. If a space $X$ is $L$-acyclic, then so is $K(\pi_n(X),n)$ for each $n$. To wh …
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2
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126
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If $\Omega f$ is an $E_\ast$-equivalence, then is $\Omega^2 f$ an $E_\ast$-equivalence?
Let $E_\ast$ be a homology theory and $f: X \to Y$ a map of spaces. Suppose that $\Omega f: \Omega X \to \Omega Y$ is an $E_\ast$-equivalence. Then is $\Omega^2 f: \Omega^2 X \to \Omega^2 Y$ an $E_\as …
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9
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451
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What is the homotopy fiber of $X \to X_{hG}$, where this is a pointed homotopy orbit?
The unpointed version is easy: the model $X = EG \times X \to (EG \times X)/G = X^{un}_{hG}$ is a fibration with fiber $G$. But when we go pointed, $X = EG_+ \wedge X \to (EG_+ \wedge X) / G = X_{hG}$ …
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6
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1
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280
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Example of a weak basic localizer which is not a basic localizer?
In Grothendieck's homotopy theory, the category $Cat$ of small categories is used to model spaces, or some localization thereof. Grothendieck gives two sets of axioms which the relevant weak equivalen …
- 64k
4
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273
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Is the mod-2 Moore spectrum a retract of a shift of its tensor square?
Let $M_p(i)$ be the mod $p^i$ Moore spectrum, i.e. the cofiber of $p^i: \mathbb S \to \mathbb S$. Upper and lower bounds on the $n$ for which $M_p(i)$ admits an $A_n$ structure are known, cf. Bhattach …
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3
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0
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108
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Examples of commutative ring spectra with graded-artinian coefficients?
Question: What are some ring spectra $E$ satisfying the following conditions?
The coefficients $E_\ast = \pi_\ast(E)$ are graded-commutative;
There is a Kunneth spectral sequence $E_\ast(X) \otimes_ …
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9
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0
answers
334
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Is an $n$-connected space a homotopy colimit of $n$-connected spheres?
A pointed, $n$-connected space $X$ admits a CW structure using just $n$-connected spheres, so $X$ is an iterated homotopy colimit of $n$-connected spheres (i.e. you start with spheres and keep taking …
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7
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If a loopspace admits space-level power operations, is is a higher loopspace?
Let $G$ be a group. Suppose that the map $G \to G$, $g \mapsto g^r$ is a group homomorphism for every $r \in \mathbb N$. Then $G$ is abelian. Is this also true homotopy-theoretically?
(In the ordinar …
- 64k
6
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2
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442
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Is higher-order excision related to higher-order cohomology operations?
Here are some things which are almost, but not quite, [representable by] (co)homology theories:
$n$-excisive functors (in the sense of Goodwillie calculus), for $n \geq 2$.
The domains and codomains …
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12
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Does every (co)homology functor (in particular, stable homotopy) factor through chain comple...
Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the …
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15
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1
answer
886
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Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences?
The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories $\mathsf{Cat …
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