Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
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Can the intermediate Chern classes be expressed as Euler classes?
General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its ...
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The homology of the orbit space
Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper).
Is there a way to understand the homology ...
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Which rings are cohomology rings?
Which rings can arise as cohomology rings of algebraic varieties?
To be more specific, take a Weil cohomology theory $H^*$ with coefficients in a field $K$ of characteristic 0 defined for smooth ...
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Kan's simplicial formula for the Whitehead product
In his article on Simplicial Homotopy Theory (Advances in Math., 6, (1971), 107 –209) Curtis quotes a formula (on page 197) for the Whitehead and Samelson products in a simplicial group $G$. The ...
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On manifolds which do not admit (smooth) actions of finite groups
Question: Assume a smooth manifold $M$ does not admit any effective smooth group actions of finite groups $G \neq 1$, does it follow that $M$ also admits no continuous effective group actions of ...
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Lie algebras vs. graph complexes
A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
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The spectral sequence of a tower of principal fibrations
Assume we have a tower of fibrations (of simplicial sets, let's say):
$$\cdots\rightarrow X_{n+1}\rightarrow X_n\rightarrow\cdots\rightarrow X_0.$$
Let $X=\lim_nX_n$ be the (homotopy) inverse limit. ...
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Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
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Rational equivalence of smooth closed manifolds
All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they ...
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Fibrations whose total spaces are more highly connected than their fibers
The (generalized) Hopf fibrations $S^1 \to S^3 \to S^2$, $S^3 \to S^7 \to S^4$ and $S^7 \to S^{15} \to S^8$ have the property that their total spaces are more highly connected than their fibers.
Are ...
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Reference request: Milnor rank of spheres
Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$...
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Progess on a Problem/Conjecture of Sullivan?
In Sullivan's postscript to his MIT notes https://www.maths.ed.ac.uk/~v1ranick/surgery/gtop.pdf he describes some problems and conjectures, where Problem 4 is: "Analyze the action of Gal($\...
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What would be the simplest analog of Langlands in algebraic topology?
It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
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Connected sum is well-defined for surfaces, proof?
EDIT: So my question is distinct from the question asked here because I am asking an easier question. Why should we have to invoke something as powerful as the "Annulus Theorem" to show that the ...
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Grothendieck 's question - any update?
This question is migrated from math.stackexchange. I ask because it is still unclear to me and I did not receive an answer.
I was reading Barry Mazur's biography and come across this part:
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