Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,852 questions
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Interpretations of the tangent Chern numbers of the complex projective spaces $CP^n$?
I've identified the generating functions for the tangent Chern numbers of the complex projective spaces $CP^n$ given in "Algebraic topology of the Lagrange inversion" by Victor Buchstaber ...
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Does a continuous variation through compact $n$ dim manifolds preserve topology?
I begin by writing the definition below that tries to capture what a continuous family /path of manifolds is. The underlying motivation behind the definition is that the transition maps should be ...
- 1,441
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Triangulated manifolds with local orientation reversal
For a (smoothly) triangulated $n$ manifold $M$, I'll say that the triangulation is amphichiral if it admits an orientation-reversing automorphism. I'll say that the triangulation is locally ...
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chain level intersection product
Let $F$ be a field and $M$ an oriented smooth $n$-manifold.
Is it always possible to construct a differential graded $F$-vector space $(C_M,d)$ equipped with a strictly coassociative coproduct $\Delta ...
- 2,165
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cellular and handle chain complex of universal covering
Let $M$ be a compact connected smooth manifold with fundamental group $\pi:=\pi_1(M)$ and universal covering $\widetilde{M}$. We can equip $M$ with a handle decomposition and we obtain the handle ...
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Goodwillie Filtration of Factorization Homology and Hochschild Homology
The factorization homology of a framed manifold $M$ with coefficient in an $E_n$-algebra $A$, $\int_M A$, is developed by Ayala and Francis. It was shown that for $M=S^1$, $\int_M A$ is the Hochschild ...
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Functoriality in exodromy
Say we have a nice topological space $X$ with two nice (finite, conical...) stratifications $A$ and $B$ such that $B$ is coarser than $A$. Then we have a natural embedding of categories of ...
- 1,247
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Orientation local system of a vector bundle
Let $p:E\to X$ be a rank $k$ real vector bundle on a paracompact space. This question is about possible definitions of the orientation local system of $E$, which should be a local system of integer ...
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Holomorphic functions from the perspective of functional analysis
Fix an open set $U\subseteq \mathbb{C}$. Let $C(U)$ denote all continuous $f:U\to\mathbb{C}$ and $H(U)$ denote all holomorphic $f:U\to\mathbb{C}$. Equip $C(U)$ with compact-open topology, and note ...
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A question on rank two vector bundles on the real two sphere
Let $k$ be the real numbers, $K$ the compex numbers and let $x,y,z$ be independent variables over $k$. Let $f:=x^2+y^2+z^2-1$ and let $A:=k[x,y,z]/(f), B:=K[x,y,z]/(f)$ and let $S:=Spec(A)$.
In a ...
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Homotopy types and acyclic relations
Say $X$, $Y$ are two compact Hausdorff topological spaces - in my mind, one can take $X, Y$ to be both included in $\mathbb{R}^d$ - of the same homotopy type, i.e there exist $f : X \to Y, g: Y \to X$ ...
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Relationship between parametric pseudo-manifolds (Gallier) and topological pseudomanifolds
I am interested in the relationship between the concept of a Parametric pseudo-manifold (PPM), as defined by Jean Gallier ,Dianna Xu, Marcelo Siqueira (e.g., in "Parametric pseudo-manifolds",...
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Obstruction to deforming a cross-section
I am currently studying the paper
Haefliger–Hirsch, “Immersions of Manifolds” (link), and I am trying to understand the proof of Theorem 1.2.
In the final step of the proof, the authors state:
“Since ...
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A variant to the Cerf-Stasiak formula (Thurston variant)
PDF of C-S formula paper here.
Thurston pers.comm.'ed in the paper that $S=T+3/7*B$, where $S$ is Stasiaks writhe (in some practical units), $T$ Taits writhe, and $B$ the checkerboard area difference (...
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Hochschild-Serre spectral sequence and an explicit filtration on the bar complex
Consider a short exact sequence of groups
$$1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1.$$
With trivial integral coefficients, the Hochschild-Serre spectral sequence in ...
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