Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,330
questions
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Unimodular intersection form of a smooth compact oriented 4-manifold with boundary
Let $X$ be a smooth compact oriented 4-manifold with nonempty boundary. Its intersection form $$ Q_X : H^2(X,\partial X;\Bbb Z)/\text{torsion}\times H^2(X,\partial X;\Bbb Z)/\text{torsion}\to \Bbb Z$$
...
2
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0
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105
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Nerve theorem for simplicial sets
There are various kinds of nerve theorems. I am wondering if the following version of nerve theorem for simplicial sets is true:
Let $X:\Delta^{\mathrm{op}}\to \mathrm{Set}$ be a simplicial set. Let $\...
0
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1
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78
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Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
We know that $M_{m \times n } $ is isomorphic to $R^{mn}$. Let's take a smooth manifold $\mathbb{M}$ in $R^{mn}$ and fix a point in $R^{n}$, say, $p$. Now realize the manifold $\mathbb{M}$ as a subset ...
2
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Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
-1
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Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
4
votes
1
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Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus? [closed]
Is automorphism on a compact group necessarily homeomorphism? I don't think so,but I think it is possible on the N-dimensional torus.
3
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1
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Why is the Vietoris–Rips complex $\operatorname{VR}(S, \epsilon)$ a subset of the Čech complex $\operatorname{Čech}(S, \epsilon\sqrt{2})$?
$\DeclareMathOperator\Cech{Čech}\DeclareMathOperator\VR{VR}$I am reading Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, and Singh - Confidence sets for persistence diagrams (see here for a version of ...
3
votes
1
answer
97
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Seifert invariants for Brieskorn manifolds $\Sigma(p,q,r)$
I've been studying Brieskorn manifolds $\Sigma(p,q,r)$ for $p,q,r\geq 2$. I know they are defined as the intersection of the complex surface $z_1^p+z_2^q+z_3^r=0$ as a subset of $\mathbb{C}^3$ and $S^...
2
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1
answer
76
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How to determine the LS category of branched covers?
Define the (normalized) Lusternik-Schnirelmann (LS) category of a space $X$, denoted $\mathsf{cat}(X)$ to be the least integer $n$ such that $X$ can be covered by $n+1$ number of open sets $U_i$ each ...
8
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Tannaka reconstruction for homotopy types
All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its ...
3
votes
1
answer
215
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Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
1
vote
0
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43
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Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
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Critical simplices of a discrete gradient vector field
I am currently reading a paper on topological data analysis which includes som discrete Morse theory arguments. I got stuck on a corollary that in the paper I'm reading is described as simply ...
3
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References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections
A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...
2
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Topological meaning of a "totally recurrent" 1d foliation in 3-manifold
I'm trying to understand Sullivan's "cycles for the dynamical study..":
https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0033.pdf
which I find very complicated being ...