Questions tagged [homotopy-theory]
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.
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Delta-generated spaces vs CW complexes
$\newcommand\Top{\mathrm{Top}}\newcommand\CW{\mathrm{CW}}\newcommand\Deltagenerated{\text{$\Delta$-generated}}\newcommand\Spaces{\mathrm{Spaces}}\newcommand\DeltaSpaces{\text{$\Delta$-Spaces}}$I am ...
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Induced map of degree $k$ self map of a sphere in the higher homotopy groups
Let $f:S^n\rightarrow S^n$ be a degree $k$ map. Then $f$ induces maps $\pi_l(S^n)\rightarrow \pi_l(S^n)$.
I believe that in the stable range ($l\leq 2n-2$) this map is multiplication by $k$. Unstably ...
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Model structure on simply-connected topological spaces in which the weak equivalences are the rational homotopy equivalences
I recently started learning rational homotopy theory, and found the claim on page 7 of this survey that there is a suitable model category structure in which the weak equivalences are the rational ...
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The induced map between push outs in an exact infinity category
Let $(\mathcal{C} , \mathcal{M} , \mathcal{E})$ be an exact $\infty$-category. (I am following the definition in Higher Segal Spaces $I$ by Dyckerhoff and Kapranov). Assume that $F$ is a cofiberation ...
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Connecting Harris's proof of periodicity to the Bott element
$\newcommand\ku{\mathrm{ku}}$In Denis Nardin's lectures on stable homotopy theory, he reproduces Bruno Harris's proof of Bott periodicity from group completion and goes on to identify the Bott element ...
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Postnikov invariant of crossed square
Is there a reference where Postnikov invariants of the classifying space of a crossed square have been computed ? I am especially interested in the computation of the third Postnikov invariant $B\...
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Freudenthal suspension homomorphism
I asked this question in MathStackExchange a couple of months ago, with little feedback, hence I try here.
The Hopf invariant $h(f)$ of a mapping $f:\mathbb S^{2m-1}\to \mathbb S^m$ is a homotopy ...
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Functoriality of infinite suspension spectrum functor on infinity groupoids!
Consider the functor $F: C \rightarrow D $ of $\infty$-groupoids. Is there any explicit proof somewhere in the literature that $\Sigma^{\infty}$ construction is functorial? I mean how do we define $\...
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Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
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Formal power series over the sphere spectrum?
In Section 11 of their paper https://arxiv.org/pdf/1802.03261, Bhatt-Morrow-Scholze discuss the polynomial algebra over the sphere spectrum. I'm wondering whether its possible to define a notion of ...
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Are the $K(n)$-local $E_n$-Adams spectral sequences isomorphic to the Adams-Novikov spectral sequences?
Let $H$ be a closed subgroup of the Morava stabilizer group $\mathbb G_n$. [Devinatz-Hopkins, Prop. 6.7] identifies the $K(n)$-local $E_n$-Adams spectral sequence for $E_n^{hH}$ as the homotopy fixed ...
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Why is $bo$ not flat?
Let $bo$ be the connective cover of the real $K$-theory spectrum $KO$. This is a ring spectrum, and so one can look at its Adams spectral sequence. Mahowald does this in "$bo$-resolutions", ...
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How much homotopy theory before higher category theory?
This is a soft question.
I am currently learning higher category theory in the context of stable homotopy theory and find many concepts more intuitive now that I have taken a basic course in algebraic ...
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Euler class in center of mod 2 Morava K-theory?
I consider Morava K-theory at the prime $p=2$ and height $n$. $K(n)^*$ is multiplicative and complex-oriented, but the multiplication is not commutative. Suppose I have a complex bundle $E$ of rank m ...
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Conceptual proof of Jacobi-like identity for Toda brackets
In the paper $p$-primary components of homotopy groups IV, Toda proved an identity for his bracket operation, which can be succinctly written as
$$[[\alpha, \beta, \gamma], \Sigma \delta, \Sigma \...
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