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Questions tagged [poincare-duality]

For questions involving or related to Poincaré duality.

Poincaré duality is an important result in homology theory, which relates homology and cohomology groups via an isomorphism.

It states that, under appropriate assumptions on a manifold $M$, $$H_{n-k}(M)\cong H^k(M)$$

Consider using this tag along with or one of the associated cohomology tags.

114 questions
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Poincaré duality for de Rham cohomology on non-compact manifolds

Let $M$ be an $n$-dimensional orientable non-compact manifold. Is there an isomorphism as follows, and if so how can we construct it? (Or can you provide a reference?) $$ H^{n-i}_{\operatorname{dR},c}(M, \mathbb R) \cong H_i(M,\mathbb R). $$ On…
Moisés
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Help with proof of noncompact Poincaré Duality by Hatcher.

I am trying to follow Hatcher's proof of Poincaré Duality on p. 248. Suppose $M$ is an $R$-oriented manifold. We have first defined $H^k_c(M;R)$ to be the direct limit of groups $H^k(M,M\setminus K;R)$ where we take some compact subset $K\subset M$.…
Jarne Renders
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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 Rham cohomology and relative cohomology. What I'd…
Dylan
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Poincaré's take on Poincaré duality before the advent of cohomology?

Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The $k$th and $n-k$th Betti numbers, $b_k$ and $b_{n-k}$ of a closed orientable n-manifold are equal. $$b_k = b_{n-k}.$$ From…
wonderich
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What is the essence of ``the naturality of the cap product''?

Associated to a continuous map $f : X → Y$, there are natural pushforward and pullback maps on homology and cohomology, respectively, denoted $f_∗ : H_∗(X) → H_∗(Y)$ and $f^*: H^* (Y) → H^* (X)$. These are related by the projection formula, also…
wonderich
  • 6,059
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Cellular diagonal approximation

Let $X$ be a Co-H space with a finite CW structure. Composing the comultiplication $c:X \rightarrow X \vee X$ with the inclusion $i:X \vee X \rightarrow X \times X$ gives a map $$i \circ c \simeq \Delta:X \rightarrow X \times X,$$ homotopic to the…
Matt
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On the proof that "Poincare dual=Thom Class"

Suppose that $S$ is an oriented smooth $s$-manifold, and $\pi :E\to S$ is an oriented real vector bundle over $S$. The Thom isomorphism asserts that the integration along the fiber defines isomorphisms$$H_{cv}^*(E)\cong H^{*-n}_{dR}(S).$$The Thom…
Ken
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Generalising dual triangulation of manifolds

We know the following “geometric” version of Poincaré duality: Let $M$ be a closed $m$-dimensional manifold and let $\mathfrak{X}_*$ be a finite simplicial complex with $|\mathfrak{X}_*|=M$. We can build a dual cell complex $\mathfrak{X}^*$: For…
FKranhold
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Proof of Poincaré duality

I am working through a proof of Poincaré duality. I don't understand the one step marked in bold. Let $R$ be a ring. Pick an $R$-orientation $(o_x; x\in\mathbb{R}^m)$ of $\mathbb{R}^m$. Pick $r\in \mathbb{N}$ and let $o_{B_r}$ be the unique…
Margaret
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Faces of the cap product

Let $X$ be a topological space and $A\subseteq X$ an open subspace. Let $R$ be an associative unital ring. Define the cap product $$\cap\colon S^q(X,A;R)\otimes S_{p+q}(X,A;R)\rightarrow S_p(X;R)$$ on singular simplices $a$ by $\beta \cap (a\otimes…
Margaret
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Poincaré duality and coefficients in the circle group

In Hatcher, Poincaré duality is stated for coefficients in a ring rather than a general abelian group. I am wondering whether it also holds when taking coefficients in the circle group, which is not a ring.
mathieu
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Closed orientable 4-manifolds with $H_2(M)\cong \mathbb{Z}$ do not admit free actions of $\mathbb{Z}/2$

The questions asks us to show that if $M$ is a closed orientable 4-manifold such that $H_2(M)$ is rank $1$, then $M$ does not admit a free action of $\mathbb{Z}/2$. My attempt has been to suppose $M$ has a free action of $\mathbb{Z}/2$. So there is…
user683708
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Hatcher's proof of Poincare Duality

Thm. 3.35, pg245. The proof begins 247-248. Hatcher defines a duality map $$D:H^k_c(M;R) \rightarrow H_{n-k}(M;R) $$ where the LHS is compactly supported cohomology. An element can be represented by $\varphi \in H^k(M,M-K:R)$, $K \subseteq M$…
Bryan Shih
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Definition of Lefschetz Number in Bott&Tu and in Gullemin&Pollack Differs by a sign?

In Bott and Tu's Differential Forms in Algebraic Topology, the Lefschetz number of a map $f:M\to M$ between an oriented compact manifold $M^m$ is defined, as in any algebraic topology text, to be $L(f)=\sum_q (-1)^q tr(f^*|_{H^q(M)})$, where we are…
Tianyi Wang
  • 425
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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*} 0\rightarrow H^1(U)\oplus H^1(V) \rightarrow…
Bigolini
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