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        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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        1answer
        111 views

        Snaith splitting for operads in spectra?

        Let $O$ be an augmented operad. Then there is a functor $J^O: Top_\ast \to Top_\ast$ which takes $X$ to the free $O$-algebra on $X$ subject to the condition that the nullary operation coincides with ...
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        2answers
        414 views

        Computing the differentials in the Adams spectral sequence

        Assume you are given an explicit presentation of the $E_2$-terms of the Adams spectral sequence. Are the differentials on $E_2$ and further algorithmically computable? I do not care how practical it ...
        4
        votes
        1answer
        120 views

        Cohomogy of local systems over CW-complexes

        Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $\dim(H^k(M,F))$ is at most the number of $k$-cells times $\operatorname{...
        6
        votes
        1answer
        442 views

        Italian-style algebraic geometry in homotopy theory?

        In homotopy theory, stacks can be occasionally useful (i.e. in the chromatic story). I come from a differential geometry background, so when people say that algebraic geometry is useful in homotopy ...
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        votes
        0answers
        65 views

        Homotopy colimits of simplicial objects

        Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...
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        2answers
        845 views

        What is the generator of $\pi_9(S^2)$?

        This is more or less the same question as [ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way not only topologists can ...
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        votes
        0answers
        84 views

        Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

        $CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
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        homotopy VS isotopy classes of embeddings

        Let $X$ any compact set in $R^n$, not necessarily a manifold. Let $f,g:X \longrightarrow M^k $ be two homotopic PL embeddings of $X$. When $X$ is an $m$-manifold then the two embeddings are also ...
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        votes
        0answers
        69 views

        Homotopy of rigid analytic spaces

        Let $K$ be a complete non-Archimedean valued field (I think the valuation does not have to be discrete). For a paracompact strictly $K$-analytic space, I have seen at least two definitions of ...
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        votes
        1answer
        156 views

        A question about the group $[HZ,KU]$

        I don't know if the following question is obvious, but can't figure it out. I want to ask if it is known what $[HZ,KU]$ is? Here $KU$ is the complex $K$-theory.
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        votes
        1answer
        152 views

        Local quotient covers for derived Deligne-Mumford geometric stacks of Toen-Vezzosi

        Let $\mathcal{X}$ be a separated Deligne-Mumford stack, and $X$ its coarse moduli space. A well-known lemma establishes an etale covering $X_{\alpha} \rightarrow X$, such that for each $\alpha$, there ...
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        votes
        1answer
        293 views

        How weird can a ring spectrum be if all of its modules are free?

        Let $R$ be a ring spectrum. If $\pi_\ast(R)$ is a graded field, then all module spectra over $R$ are free. But I don't believe the converse holds. How badly can it fail? I'm assuming that $R$ is at ...
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        votes
        0answers
        163 views

        Examples and non-examples of Tannakian $\infty$-categories

        My definition of a Tannakian $\infty$-category is taken from this paper ("Tannaka duality over ring spectra" by James Wallbridge). (p. $53$, definition $7.9$) Let $R$ be an $E_{\infty}$-ring, $C$ a ...
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        1answer
        131 views

        A question about maps of spectra

        Maybe this is obvious but I can′t figure it out. Suppose that we have a connective spectrum $X$ and consider a spectrum $Y$. Let $Y^{\prime}\to Y$ be its (-1)-connective cover. Is it true that $[X,...
        15
        votes
        1answer
        448 views

        What is classified by generalised Eilenberg MacLane spaces?

        Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$. In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the ...

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