Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,773 questions
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Orbits in equivariant stable homotopy
Let $G$ be a finite group (probably all relevant phenomena occur already when $|G|=2$). Let $\text{Ab}$ be the category of sets, let $G\text{Ab}$ be the category of abelian groups wit $G$-action. ...
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A question of Steenrod
Let $G$ be a finite group and $A$ a torsion-free $\mathbb{Z}[G]$-module. Fix $n>0$.
Steenrod posed the question (see Sect. 3 in the paper of Swan below) whether there exists a finite $G$-CW complex ...
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What physical theories use mathematical discipline with a high level of abstraction? [closed]
I know several classic examples of modern physical theories, such as quantum field theory (I mainly study topological field theory), as well as Yang-Mills theory and research on the existence of ...
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Is there a relative version of the s-cobordism theorem?
In Smale's paper "On the Structure of Manifolds", he proves a "relative" version of the h-cobordism theorem:
Theorem (Smale): For $i \in \{1,2\}$, let $M_i$ be a closed oriented ...
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The first Pontrjagin class of Milnor's example of exotic 7-sphere
In the construction of exotic 7-spheres, Milnor used the fact the first Pontrjagin class $p_1(\xi_{ij})$ is linear in $i,j$. He claims that this is clear but I don't feel that. I tried to find some ...
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How to deform the geometric realization of a precubical set as directed space to put a point in the middle of a cube?
Consider a $n$-cube $[0,1]^n$ equipped with the standard directed space structure in Grandis' sense: the directed paths are the continuous paths which are locally non-decreasing with respect to each ...
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Brown-Comenetz type dual as a target for discrete invertible field theories?
Freed and Hopkins use the spectrum $I\mathbb C^\times$ to define the target of discrete invertible field theories. I have trouble understanding, how this arises. Naively, physically, I would think ...
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Polynomial conditions on cycle space of simple graph
Let $G=(V,E)$ be a simple cubic graph and $\Phi:=\lbrace \phi\rbrace$ a set of cycles such that for every edge $e\in E$ there is a $\phi\in \Phi$ such that $e\in \phi$. We say that $\Phi$ covers $G$. ...
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Correspondence local systems & $\Bbb Z \pi_1(A)$-modules & compatibility issues
Let $X$ be a topological space and $G \subset \text{Aut}(X)_{\text{top}}$ a finite group acting faithfully on $X$ "nicely enough", where "nice" = we can form categorical quotient $...
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Recursive pointfree approach to algebraic topology
$\newcommand\seq[1]{\langle#1\rangle}$A large number of important topological results require simplicial-algebraic machinery (or comparable) to prove. This machinery is ingenious, impressively so even,...
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Minimal CW complex detecting all powers of euler class
Let $n \in \mathbb{Z}_{>0}$. I'm wondering about pairs $(X,V)$ where $V$ is a $2n$-dimensional vector bundle on the CW complex $X$, such that
The ring homomorphism $\mathbb{Q}[e] \to H^*(X;\...
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Are two symplectic fibration(resp. Hamiltonian fibration) are smoothly fibration isomorphic if it holds continuously?
Let $P, P'$ be symplectic(resp. Hamiltonian) fibrations over a base $B$.
If two $P, P'$ are continuously symplectic fibration(resp. Hamiltonian fibration) isomorphic, then are they also smoothly ...
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Fundamental class from smooth atlas
Let $M$ be an orientable $n$-dimensional manifold with a smooth oriented atlas $A :=\{(U_\alpha,\Psi_\alpha)\}_{\alpha}$ so that every non-empty $U_\alpha\cap U_\beta$ is contractible.
Then we can ...
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Finite type subcomplex which is homology equivalent to the given one
Let me call a simplicial set $Y$ to be of finite type, if there are finitely many simplices in each degree.
Suppose now $T$ is a finite CW complex and $X$ is arbitrary simplicial set whose homotopy ...
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Different notions of Topological Triangulated Categories
In Schwede 2008 a topological triangulated category is defined as a triangulated category equivalent to a full triangulated subcategory of the homotopy category of a stable model category.
In Schwede ...
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