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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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      4
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      1answer
      75 views

      Group completion of $E_k$-algebras

      Let $X$ be an $E_k$-algebra. We can form the delooping $BX$, which is a $E_{k-1}$-algebra. The space $\Omega B X$ is again an $E_k$-algebra, which is grouplike (i. e. $\pi_0(\Omega B X)=\pi_1(B X)$ is ...
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      166 views

      Are the real and complex Adams operations compatible under the inclusions $U(n) \rightarrow SO(2n)$?

      Does the following diagram commute? $$ \require{AMScd} \begin{CD} BU @>{\psi^k}>> BU \\ @VVV @VVV \\ BO @>{\psi^k}>> BO \end{CD} $$ Evidence for: $rc = 2$, it works for $BU(1) \...
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      Weak homotopy equivalence of sites

      There are several notions of weak homotopy equivalence for topological spaces. The standard one can be formulated as follows: a map of spaces $X\to Y$ is a homotopy equivalence if the map of ...
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      62 views

      Cyclic homotopies of quotients of $S^3$

      We are given a free action of an abelian finite group on $S^3$. Let $L$ denote the quotient space and let an element $\alpha \in \pi_1 L =G$ be given. Does there exist a cyclic homotopy $h_t:L \to L$ ...
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      Homotopy pullback of motivic weak equivalences

      How to see that whether $\mathbb{A}_1$-weak equivalence is closed under homotopy pullback? Let $L_{Nis}(Sm_S)$ be the Nisnevich localization, how to compute the homotopy pullback of maps $\mathbb{A}_1\...
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      260 views

      In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?

      I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also? Given the broad scope of this question I ...
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      votes
      1answer
      159 views

      In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?

      In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also? Note that ...
      31
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      3answers
      2k views

      In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

      I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces. If it is true that: In a Topological Space, if there exists a loop that cannot ...
      6
      votes
      1answer
      187 views

      Why does every chain complex have a map into its cone?

      In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...
      2
      votes
      1answer
      200 views

      Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

      Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
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      62 views

      Stable splitting of products

      This question concerns the well-known homotopy equivalence $$ \Sigma (X\times Y) \simeq \Sigma (X \vee \ Y) \vee \Sigma (X\wedge Y) $$ (I'm happy to use only CW complexes). I can see that there is ...
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      1answer
      238 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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      210 views

      $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

      First of all I want to apologize for the much too long post. A Lie group $G$ is acting on a smooth manifold $M$, then we define \begin{align*} T^k_G(M)= (S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
      6
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      1answer
      120 views

      Diffeomorphism type of the added sphere in simply connected surgery

      A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...
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      0answers
      192 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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