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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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### 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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### 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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### 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 ...
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### 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 ...
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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 $Cone(C)=\left(C_{i-1} \oplus ... 1answer 200 views ### Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers Let$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 ... 0answers 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 ... 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 ... 0answers 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$M, then we define \begin{align*} T^k_G(M)= (S^\bullet \mathfrak{g}\otimes T^k_\mathrm{... 1answer 120 views ### Diffeomorphism type of the added sphere in simply connected surgery A classical result of simply connected surgery theory, is that if two normal mapsf:M_i\rightarrow X$,$i=0,1$are normally cobordant and if the dimension of the manifolds is odd, there exists a ... 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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