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        Questions tagged [at.algebraic-topology]

        Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

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        44 views

        Higher K-theory groups and Fredholm operators

        As is well known, whenever $X$ is a compacta space, the Atiyah-Janich theorem says that there is an identification $$[X,\mbox{Fred}(H) ]\cong K^0(X) $$ between the set of homotopy classes of maps from ...
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        1answer
        144 views

        A question on relative equivariant cohomology

        Suppose that we have defined an extraordinary $G$-equivariant cohomology theory $H$ (say $G$ is a compact group). If $X$ is a $G$-space and $A\subset X$ is a closed $G$-equivarant contractible ...
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        78 views

        Minimal cell structures

        Is there some reasonable obstruction theory which detects whether a finite type complex can be provided with a cell structure such that number of $i$-cells is equal to the maximal rank of $i$th ...
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        1answer
        194 views

        Equivalent definitions of Thom spectra

        Background and notations: Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...
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        0answers
        133 views

        Question about fundamental group of complements of divisors

        Let $X$ be a quasi-projective smooth algebraic variety (over $\mathbb{C}$) and $D$ an irreducible divisor, such that $(X,D)$ is log smooth. Is it true that there is an exact sequence $\mathbb{Z}\...
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        1answer
        197 views

        Reference request: mod 2 cohomology of periodic KO theory

        The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $...
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        75 views

        Simplicial sets of categories as models for $(\infty,1)$-categories [duplicate]

        Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise. In my understanding, there are several models for $(\infty,...
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        1answer
        114 views

        Strøm model structure on nonnegatively graded chain complexes

        Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes. The ...
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        1answer
        303 views

        Oriented vector bundle with odd-dimensional fibers

        Is it true that for every oriented vector bundle with odd-dimensional fibers, there is always a global section that vanishes nowhere?
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        0answers
        188 views

        Homotopy equivalence of $K$-theory and $G$-theory

        Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...
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        5answers
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        What is the intuition for higher homotopy groups not vanishing?

        The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This ...
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        1answer
        285 views

        Replacing the Fibre of a Fibration

        This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature. Let $p:E\rightarrow ...
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        1answer
        151 views

        Petries exotic circle action

        In the paper "S^1-actions on homotopy complex projective spaces" by Petrie (Bulletin of the AMS, 1972), Petrie constructs a smooth circle action on $\mathbb{CP}^{3}$ (page 148). The fixed point set ...
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        1answer
        399 views

        Proj construction in derived algebraic geometry

        The question My question is easy to state: Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”? Given the vagueness of the question, you’...
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        2answers
        227 views

        Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid

        Let $X$ be a path-connected smooth manifold, it is known that: $$H^1(X):=H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R).$$ Explicitly, a closed one-form $\alpha$ gives a function on $\pi_1(X)$ by $[\...

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