# Questions tagged [poincare-duality]

For questions involving or related to Poincaré duality.

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• 395
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### Why is the signature of a manifold homotopy invariant?

I'm going through some applications of Poincaré duality for de Rham cohomology in Greub's Connections, Curvature and Cohomology. I was wondering about the homotopy invariance of the signature of a ...
• 101
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### Intersection of submanifolds, cup products, and Poincaré duality

Recently I have been thinking and inquiring about how "cup products are dual to intersection of submanifolds", and wanted to verify whether the following is accurate (and to find a source ...
• 404
1 vote
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### An example of a non hausdorff space

Here is the question I am trying to understand its solution: Show that there exist nonorientable $1$-dimensional manifold if the Hausdorff condition is dropped from the definition of a manifold. I ...
1 vote
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### Let $A$ be a 2-torsion finite abelian group . Let $f: A\to (\Bbb{Q}/\Bbb{Z})^n$ be an arbitrary group homomorphism. $image(f)\le 2^n$?

Fix a natural number $n$. Let $A$ be a 2-torsion finite abelian group . Let $f: A\to (\Bbb{Q}/\Bbb{Z})^n$ be an arbitrary group homomorphism. How can I prove $image(f)\le 2^n$? My try: From structure ...
• 6,351
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### The inverse map of the Poincare duality map

Let $M$ be an oriented compact smooth manifold of dimension $n$. Let $[M]$ be the fundamental class of $M$, that is, $[M]\in H_n(M, \mathbb Z)$. Then, the Poincare duality map is the isomorphism given ...
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• 33
1 vote
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### How Bockstein homomorphism maps under Poincarè duality?

Given a manifold $X$ and short exact sequence of abelian groups $$1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1$$ we get the Bockstein map in cohomology ...
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### How to construct the action on chains of the Bockstein homomorphism for homology

Given a manifold $X$ and short exact sequence of abelian groups $$1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1$$ we get the Bockstein map in cohomology ...
1 vote
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### Relative Intersection Pairing (Poincaré-Lefschetz duality)

Let $M$ be a $2$-dimensional orientable compact manifold. (For example, a Riemann surface.) We have an intersection pairing $$H_1(X,\def\Z{\mathbb{Z}}\Z) \times H_1(X,\Z)\to \Z$$ which is unimodular, ...
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### Mayer-Vietoris Sequence and Poincare dual

Given a $4$-dimensional simply connected manifold $M$ and open sets $U,V\subseteq M$ such that $U\cup V=M$ we can compute the deRham cohomology in terms of the Mayer-Vietoris sequence: \begin{align*} ...
• 101
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### Euler characteristic of odd dimensional manifold - Hatcher

I ran into some trouble while reading through Hatcher's proof of the following: Corollary 3.37. A closed manifold of odd dimension has Euler characteristic zero. There is only one part of the proof ...
• 2,541
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### algebraic intersection number and Poincare duality

I am trying to understand the algebraic intersection number in terms of Poincare dual and the cup product. This is: Let $M$ be a compact oriented $m$-dimensional smooth manifold together with a ...
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### Poincaré-Bendixson theorem in $\mathbb{R}^{1}$

The theorem of Poincaré-Bendixson states that: For an open set $U \subset \mathbb{R}^{2}$ and a continuously differentiable vector field $F:U \rightarrow \mathbb{R}^2$ and a compact set $K \subset U$, ...
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### Sources on twisted/local (co)homology

Any suggestions for sources to learn about (co)homology with twisted/local coefficients? I've read Hatcher's section 3.H and have previously been directed to Sakasai's paper A Survey of Magnus ...
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### Analogous of Poincaré Duality for relative homology and relative cohomology

I am studying Morse Theory on finite dimensional and compact manifolds using homology groups and relative homology groups on $\mathbb{Z}$. I want to show that this theory could be developed using De ...
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