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        Stack Exchange Network

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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 ...

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        Spectra with “finite” homology and homotopy

        As known, any non-trivial finite spectrum $X$ can not have non-zero homotopy groups $\pi_i(X)$ only for finite number of $i$. As I understand, the same is true for any spectrum $X$ with finitely ...
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        1answer
        149 views

        HKR generalized character theory question regarding a certain construction

        In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology ...
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        vote
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        34 views

        Maps having the right lifting property against cofibrations of compact spaces

        I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...
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        130 views

        Homotopy functor calculus vs functor calculus in additive categories

        Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance). Then ...
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        2answers
        301 views

        Is there any significance to Bousfield localization in the non-derived context?

        The term "Bousfield localization" of a category $C$ is used in roughly two different ways: There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
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        213 views

        How to define 0-sphere in a category with zero object?

        The 0-sphere $S^0$ is the coproduct of two points, $$S^0 \simeq \ast \coprod \ast$$ How to define 0-sphere in a category with zero object? Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, ...
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        votes
        2answers
        272 views

        Maps from 2-Torus to SO(3)

        Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]
        3
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        168 views

        Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

        Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
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        2answers
        303 views

        Do finite simplicial sets jointly detect isomorphisms in the homotopy category? [duplicate]

        Let $\mathcal{H}$ denote the homotopy category associated with the Kan-Quillen model structure on $\mathbf{sSet}$. Suppose we have a map $f\colon X \to Y$ between Kan complexes, such that for every ...
        8
        votes
        2answers
        419 views

        Sphere spectrum, Character dual and Anderson dual

        The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. However, could you help me to appreciate the mathematical meanings of the following: What is the significance of ...
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        75 views

        Axiomatization of the shuffle decomposition

        I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state ...
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        1answer
        241 views

        Commutativity up to homotopy implies strict commutativity, for lifting problems

        Suppose we have a commutative diagram $\require{AMScd}$ \begin{CD} A @>>> X \\ @VVV & @VVV \\ W @>>> Y\\ \end{CD} where the map $A\rightarrow W$ is a cofibration and the ...
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        1answer
        286 views

        Milnor Conjecture on Lie groups for Morava K-theory

        A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...
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        0answers
        338 views

        Questions about obstruction theory (Hatcher's book)

        I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
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        1answer
        193 views

        A question on eversion of (odd) spheres

        At the right column of the page 654 of the paper, R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of ...

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