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        Questions tagged [at.algebraic-topology]

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

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        votes
        0answers
        8 views

        Locally trivial fibration over a suspension

        For $X$ a paracompact space, I am trying to classify all locally trivial fibration, with fiber-type a space $F$ such that $G_F = Homeo(F)$ with the C.O. topology is a topological group, and base the ...
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        1answer
        20 views

        homology of the fiber

        Let $f:X\rightarrow Y $ be a fibration (with fiber $F$) between simply connected spaces such that $H_{\ast}(f):H_{\ast}(X,\mathbb{Z})\rightarrow H_{\ast}(Y,\mathbb{Z})$ is an isomorphism for $\ast\...
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        vote
        1answer
        80 views

        $X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a Co-H-Space

        I have asked the below question on MathSE (with a 200 point bounty) but have yet to receive an answer there, and so am trying here. I am happy to remove it if it is nevertheless decided that this ...
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        votes
        2answers
        172 views

        What is the total square on the dual Steenrod algebra?

        The dual Steenrod algebra ($p=2$) has generators $\xi_n$ and these have conjugates that are often labeled $\zeta_n$. I am curious about the left and right actions of the Steenrod algebra on its dual, ...
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        votes
        0answers
        269 views

        When is a map of topological spaces homotopy equivalent to an algebraic map?

        My question is simple, but I don't expect there are any simple answers. Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....
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        166 views

        Higher homotopy groups of Calabi-Yaus

        Is something known about the higher homotopy groups of Calabi-Yau threefolds? For example, one of the easiest CYs is the quintic, defined as the anticanonical divisor in $\mathbb{CP}_4$. What are its ...
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        vote
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        60 views

        Homotopy classes of maps between special unitary Lie group. Correction [duplicate]

        An hour ago I asked a question (under the same title) but I used a wrong notation. Here is the improved version. We consider a special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and ...
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        0answers
        71 views

        Action of the symmetric group on connected sums of manifolds (minus a disk)

        Let $M$ be a connected compact topological $n$-dimensional manifold without a boundary and with a CW-structure $M= \bigcup M^i$. We have that $$ (\#^g M)\smallsetminus D^n \simeq \bigvee_{i=1}^gM^{n-...
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        votes
        1answer
        185 views

        Homotopy classes of maps between special unitary Lie groups

        I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now. We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...
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        votes
        2answers
        627 views

        Examples of odd-dimensional manifolds that do not admit contact structure

        I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure. Can someone provide me with some examples?
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        vote
        1answer
        173 views

        Homotopy of group actions

        Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...
        12
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        1answer
        294 views

        Counter-example to the existence of left Bousfield localization of combinatorial model category

        Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists ...
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        votes
        0answers
        49 views

        Generators of the fundamental group of the solid torus [migrated]

        I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 \in \mathbb{Z} \cong \pi_1(T)$, that is the curve represents ...
        6
        votes
        1answer
        207 views

        Is the Thomason model structure the optimal realization of Grothendieck's vision?

        In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...
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        votes
        0answers
        202 views

        Topological data of $K3\times T^{2}$

        I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold . EDIT:...

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