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Questions tagged [poincare-duality]

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Poincare Inequality Involving Plancherel

I need help in proving this question: Suppose the function $f$ is differentiable and $\frac{1}{2}$-periodic (i.e., more than just 1-periodic). Show that the Poincare inequality holds with a better ...
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Poincare dual simplicial structure of complexes homotopy equivalent to manifolds

Given a closed $n$-manifold $M$, Poincare duality equips us with an isomorphism: $$H_k(M)\cong H^{n-k}(M)$$ Here I'm speaking of singular homology with coeffecient in $\mathbb{Z}_2$. Suppose now $M$ ...
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Is every 3-dimensional Poincaré complex a 3-dimensional topological variety?

I have this question, if every 3-dimensional Poincaré complex is a 3-dimensional topological manifold? Definition (Poincaré complex) $X$ is a n-dimensional Poincaré complex if $X$ have the same ...
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Regarding the proof of Poincare Duality via 5-Lemma (commutativity?)

The proof of Poincare Lemma for oriented manifolds with finite good cover (terminology of Bott, Tu) states that \begin{align} H^q (M) \simeq H^{n-q} _c (M) \end{align} where $n=\dim M$. The proof ...
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singular cohomology and Poincaré duality

Suppose $M$ is a n-dimensional, finite type, oriented, smooth manifold. A $k$-dimensional cycle in $M$ is a pair ($S$,$\phi$), where $S$ is a compact, oriented $k$-dimensional manifold without ...
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Intersection number and cup product

It is known that for a closed oriented smooth manifold $M$, if A and B are oriented submanifolds of $M$, and if A and B intersect transversely, then the Poincare dual of A ∩ B is the cup product of ...
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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 ...
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Poincaré duality and quantum (co)homology for $S^2 \times S^2$

Consider the symplectic manifold $(M,\omega):=(S^2 \times S^2, \omega_{FS}\oplus \omega_{FS})$. The homology $H_*(M;\mathbb{C})$ has as a basis the 4 non-trivial classes: $[pt],[M],A$ and $B$, where $...
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Projection formula for proper maps of manifolds

Let $X,Y$ be (not necessarily compact) orientable manifolds without boundaries of dimension $m$ and $n$. Let $f : X \rightarrow Y$ be a proper map. Assume that all the cohomologies have coefficients ...
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Alexander duality on (co)chain level

In the book by Stöcker & Zieschang, the Poincaré duality is obtained by an isomorphism $\gamma \colon \operatorname{Hom}(C_q(K), \mathbb{Z}) \to C_{n-q}^{\ast}(K^{\prime})$, $\gamma(\varphi) = \...
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When compactly supported cohomology ring is zero?

The compactly supported cohomology (over rational) of a compact manifold is the same as ordinary cohomology. Also, we can relate the compactly supported cohomology of the oriented non-compact manifold ...
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Simplicial intersection product

Let $X$ be both a simplicial set and a closed $n$-dimensional manifold. We have a duality isomorphism $H^k(X)\to H_{n-k}(X)$. Furthermore, let $Y,Y'\subseteq X$ be two subcomplexes and closed ...
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Relation between cup and cap product

I am trying to prove that a certain diagram involving cup and cap products commutes, but there is a step which I don't understand. Let $X$ be a topological space and $R$ a commutative ring. If I fix ...
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Theorem about Poincaré algebra

I read about Cohomology ring of finite-dimensional Grassmanian, and author used algebraic lemma as "every algebraist know about it, it is very easy". I've tried to prove it myself, search through the ...
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Presentation for $H^*(\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1)$

Let me preface this with saying that I'm not a math student, but rather, a physics student and I'm trying to learn algebraic topology on my own. I would like to determine the following: Let $H_*(\...
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How to show the following diagram commutes?

In the proof of Lemma 3.5 in 3-Manifold Topology, the commutativity (up to a sign) of the following diagram is utilized: How to show the commutativity? In fact the diagram commutes up to a sign, ...
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Projective space and sections inducing the same homology morphisms

Let $X$ be a space and let $\jmath:X\times \mathbb{C}P^1\to X\times \mathbb{C}P^h$ be the obvious inclusion. Then Künneth gives us an isomorphism $\jmath^*:H^2(X\times\mathbb{C}P^h)\to H^2(X\times \...
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On the proof of Poincaré dual of transversal intersection

I'm studying a proof of the following: Let $M$ be an oriented manifold of finite type without boundary. Let $R,S$ be two transversal embedded submanifolds without boundary; if $\eta_{R}, \eta_{S}, \...
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How should I understand fundamental class of diagonal of product space

I have some difficulty understanding the fundamental class of diagonal of product space of compact oriented manifold $\Delta \subset M \times M$. For example, when $M = S^1$, the generator of $H^1(S^1 ...
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Cup products in compactly supported and twisted cohomology of manifold.

I know very well about the cup product in ordinary cohomology. My question is that how we compute the different cup product with twisted or compactly supported cohomology i.e, $$\cup:H^{*}(M,\mathbb{...
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Relation between twisted and untwisted rational cohomology of non oriented manifold.

If manifold is closed oriented, we have Poincare duality to relate homology and cohomology groups of manifold. My question is that what is the Relation between twisted and untwisted rational homology ...
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Explicit description of Poincare dual of graph?

Suppose $M^m$ and $N^n$ are smooth, connected, closed, oriented manifolds, with $m\ge n$.If we have a smooth map $f:M\rightarrow N$, we can define a graph as the submanifold $\Gamma=\{(x, f(x))\mid x\...
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Hatcher Corollary 3.39, making unstated assumptions?

The Corollary 3.39 on page 250 says: If $M$ is a closed connected orientable $n$-manifold, then an element $\alpha \in H^k(M;\mathbb{Z})$ generates an infinite cyclic summand of $H^k(M;\mathbb{Z})$ ...