# Questions tagged [poincare-duality]

The poincare-duality tag has no usage guidance.

15
questions

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### Relative, local coefficient Poincaré duality

Let $X$ be a connected, oriented and aspherical (meaning $\pi_i(X) = 0$ for $i\geq 2$) manifold of dimension $n$, and $S$ a local coefficient system (i.e., a $\pi_1(X)$-module). If $X$ is closed, we ...

3
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0
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### Reference request: inverse image in singular homology as in Chow groups

I come from algebraic geometry and I have trouble finding a reference to check the construction of the inverse image in singular homology, analogous to that of the Chow groups. Let me be more precise:
...

3
votes

1
answer

177
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### Is the cap-product map injective for singular varieties?

Let $X$ be a singular, projective (complex) variety of dimension $n$ with at worst isolated singularities. We know that taking cup-product with the cohomology class of $X$ induces a morphism from the ...

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0
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123
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### Duality in Spc with ∧ and [-,-]

I am thinking about two duality theorems for H-spaces and their actions.
By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider ...

1
vote

1
answer

93
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### Simplicial cochain representing the pullback of a class Poincaré dual of a submanifold

Let $K$ be a simplicial complex of dimension $n$, $M$ be a topological manifold, and $f \colon |K|
\to M$ be a continuous map. Let $X$ be an embedded manifold in $M$ of codimension $n$, such that
$f(|...

5
votes

1
answer

545
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### Betti numbers of non-orientable $3$-manifolds

Let $M^3$ be a compact $3$-manifold with boundary $\partial M$.
If $M$ is orientable, then it is known (see Lemma 3.5 here) that $2\dim(\ker(H_1(\partial M,\mathbb{Q})\rightarrow H_1(M,\mathbb{Q})))=\...

3
votes

1
answer

320
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### Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.
Consider the natural map
$$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{...

1
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1
answer

222
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### Confusion about relative Poincaré duality in the context of $\ell$-adic cohomology

I have recently learned about relative Poincaré duality in the book Weil conjectures, perverse sheaves and $\ell$-adic Fourier transform by Kiehl and Weissauer (2001). The reference is section II.7. ...

2
votes

1
answer

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### Composition of Gysin and restriction maps on $\ell$-adic cohomology

I already posted this question on mathstackexchange there, but I figured that it may have more replies here.
I follow the notations of Milne's lectures notes on etale cohomology, most specifically ...

5
votes

1
answer

231
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### Gysin isomorphism in de Rham cohomology using currents

I'd like to find a reference for the following fact.
First, some background: we can define de Rham cohomology of a smooth manifold $X$ of dimension $d$ using the de Rham complex
$$
\Omega^0_X\to \...

1
vote

0
answers

74
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### Does Poincaré duality link topological study and representation study of a given Lie group?

The Poincaré duality for an oriented n-manifold M takes the form : $$H^\star(M) \simeq H_c^{n-\star}(M)^\vee.$$
Instead of M take now a real Lie group G. We can basically study it by looking at its ...

2
votes

0
answers

122
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### Triple insersection number of a surface in three-manifolds

I heard something about the triple intersection number $\text{mod}(2)$ (but maybe also $\text{mod}(n)$) of a surface in an orientable three-manifold but I couldn't find a precise definition. My guess ...

0
votes

0
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67
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### Proving an equality of differential forms by assuming some perhaps topological condition

Let say I want to show two differential forms $\omega_1$ and $\omega_2$ on a smooth manifold $M$ are equal. Of course it suffices to show $\omega_1=\omega_2$ locally, i.e. the equality holds over ...

3
votes

2
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655
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### Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism 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 ...

2
votes

0
answers

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### Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance:
https://math.stackexchange.com/a/14469/454016
Poincaré duality is explained through a duality of ...