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      Questions tagged [at.algebraic-topology]

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

      3
      votes
      1answer
      54 views

      Zero surgery on a Seifert fiber space

      I have a problem with understanding what is a neighbourhood of a singular fiber in a Seifert fibered space coming from the zero surgery. For me a 3-manifold $Y$ is a SFS if it has a decomposition into ...
      6
      votes
      0answers
      115 views

      Use of Steenrod's Higher Cup product and the graded-commutativity

      In Steenrod's ``Products of Cocycles and Extensions of Mappings (1947),'' which derives [Theorem 5.1] $$ \delta(u\cup_{i} v)=(-1)^{p+q-i}u\cup_{i-1}v+(-1)^{pq+p+q}v\cup_{i-1}u+\delta u\cup_{i}v+(...
      1
      vote
      1answer
      161 views

      Non-abelian cohomologies

      Let A be a non-commutative algebra and let X be some geometric space (such as a topological space or an algebraic variety or scheme). Is there a notion of cohomology ring of X with coefficients in A? ...
      2
      votes
      0answers
      62 views

      Why is a homotopy limit of a cosimplicial space not the ordinary limit?

      I've been trying to compute a homotopy limit of a cosimplicial object $X: \Delta \to \mathscr{M}$, where $\mathscr{M}$ is some simplicial model category, we may take it to be spaces. I'm wondering ...
      7
      votes
      2answers
      332 views

      What are the advantages of simplicial model categories over non-simplicial ones?

      Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
      1
      vote
      0answers
      63 views

      The table reduction morphism of operads from Barratt-Eccles to Surjection

      The Barratt-Eccles operad $E$ in the category of simplicial sets is obtained by applying the nerve functor to the canonical operad $\{\Sigma_n\}_{n>0}$ in groups. Berger-Fresse defined here an ...
      1
      vote
      0answers
      89 views

      Fibre transfer of $\mathbb{S}^1$-bundles

      Let $p:E\to X$ and $p':E'\to X'$ be two orientable $\mathbb{S}^1$-bundles. Denote their homological transfers by $p_!:H_*(X)\to H_{*+1}(E)$ resp. $p'_!$. Now let $(u,f)$ be a bundle morphism ($u:E\to ...
      2
      votes
      0answers
      99 views

      Iterated free infinite loop spaces

      Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted ...
      5
      votes
      1answer
      97 views

      Two models for the classifying space of a subgroup via the geometric bar construction

      Let $H$ be a topological group which is a subgroup of two other topological groups $G$ and $G'$. It follows (from Rmk 8.9 in May - Classifying spaces and fibrations (MSN, free)) that there exist weak ...
      1
      vote
      0answers
      28 views

      $\omega$-nilpotent cover of a recurrent surface

      Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville. $\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
      2
      votes
      0answers
      37 views

      Tensor product of an L-infinity algebra with the cochains on the 1-simplex

      I would like to understand the $L_\infty$ structure on the tensor product of an $L_\infty$ algebra (over $\mathbb{R}$) $L$ with the normalized cochains on the one-simplex $N^*(\Delta^1)$. This latter ...
      3
      votes
      0answers
      55 views

      Globalising fibrations by schedules

      In a paper in Fund Math 130.2 (1988): 125-136. http://eudml.org/doc/211719 Dyer and Eilenberg give an account of the local-global theorem for fibrations by proving a "Schedule Theorem" that, given a ...
      4
      votes
      0answers
      105 views

      References on $HZR$ theory

      Are there references available on $HZR$ theory ? I found on ncatlab that this is "genuinely $\mathbb Z/ 2\mathbb{Z}$-equivariant cohomology version of ordinary cohomology" Found nothing on wikipedia,...
      3
      votes
      1answer
      200 views

      Divisibility of a divisor

      Let $X$ be a smooth complex projective curve and $f \colon X \to Y$ an étale Galois cover, whose Galois group $G$ is finite and of order $r$. For any $g \in G$, define $$\Delta_g = \{(x, \, g \cdot x) ...
      8
      votes
      0answers
      162 views

      $\Gamma$-sets vs $\Gamma$-spaces

      I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set. For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...

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