Questions tagged [differential-topology]
Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
7,647 questions
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Chern class of tangent bundle of hypersurface of CP^n viewed as complex of coherent sheaves
I would like to compute the total Chern class of the tangent bundle to a hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ by viewing the following short exact sequence as a complex of coherent sheaves ...
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Book Recommendation for Vector Bundles
I am interested in learning more about general vector bundle theory. More specifically, vector bundles of class $C^k$ for $k\in\mathbb{N}$ or $C^\infty$ or real-analytic whose fibers can be given the ...
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Descend constant vector fields on Euclidean $4g$-gon via opposite-side identification to vector fields with only one singularity on genus-$g$ surface
For the case of genus $g=2$, to construct a genus 2 surface we can identify the diametrically opposed edges of an octagon:
The Construction:
Consider the regular octagon in the complex plane with the ...
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Connected sum of oriented manifolds
Given an oriented manifold $M^n$ , we consider connected sum
$$ M \# \bar{M} :=( M \setminus int \mathbf{ D^n}) \cup ( \bar{M} \setminus int \mathbf{ D^n})$$
where $\bar{M}$ is the manifold $M$ with ...
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Topological Representations of Bipartite Qubit States from a new perspective
I am studying a recent paper (MDPI Information, 2025) that proposes a topological representation of bipartite qubit states.
In the topological framework proposed, measurement outcomes are obtained by ...
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Number of critical points of a homological sphere with dimension three
Let $M$ be a homological sphere of dimension 3 with a non-trivial fundamental group and $f:M \to \mathbb{R}$ a Morse function. I need to prove that $f$ has at least six critical points.
By Hurewicz ...
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The Homotopy and Homology type of Hyperplane arrangements
This is a theorem in "Homotopy Types of Subspace Arrangements
via Diagrams of Spaces" by Ziegler and Zivaljevic. I would be interested in if we can say more in the Case where $\mathcal{A}$ ...
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Is a compact hypersurface in euclidean space orientable? [closed]
I'm trying to prove this exercise from G&P book, but I don't know if I'm right in my sketch: here it follows
By the smooth Jordan--Brouwer Separation Theorem, $\mathbb{R}^n \setminus \Sigma$ has ...
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Induced connection on pull-back of conjugate bundle
Suppose X be a Riemann surface endowed with anti-holomorphic involution $\sigma_X$. $V$ is a holomorphic vector bundle on $X$ with holomorphic connection $D$. It is a fact that $\sigma_X^*\overline{V}$...
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Minimum degree of a holomorphic map from an algebraic curve of genus g to the Riemann sphere
This is a problem I found on the Rick Miranda's book.
Problem
What is the minimum integer $k$ such that for every curve $X$ of a fixed genus $g$ there is a holomorphic map $F: X \rightarrow \mathbb{P}^...
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Maximal tubular neighborhood and the complement’s topology [closed]
Basic Settings
Let $M$ be a smoothly embedded submanifold of a smooth manifold $N$. Consider the collection of open neighborhoods $U \subseteq N$ of $M$ for which there exists a retraction $r : U \to ...
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On notational conventions between Bott & Tu Vs. Lee for differential forms
Since I have been introduced to differential forms, I have seen (naively speaking) when you apply the exterior derivative, you "wedge" together one additional $d$ of the variable in question ...
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Why are these two notions of Affine Connections equivalent?
There are two definitions of the affine connection on the tangent bundle of a $\mathcal{C}^\infty$ manifold $M$. One being in terms of a Differential Operator i.e.
Definition: An Affine Connection is ...
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Change of coordinates in the cotangent space
I am studying differentiable manifolds and I came across the definition of cotangent space. I have a doubt on how we change coordinates in the cotangent space. Let $(A,\varphi)$ and $(B,\psi)$ be ...
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Why is Morse homology isomorphic to singular homology?
The Morse-Witten complex is defined as the complex whose chain groups are generated by the critical points and the boundary map defined on generators by $\partial_k(x)=\sum_{y\in Crit_{k-1}}n(x,y)y$, ...
