Questions tagged [poincare-duality]
For questions involving or related to Poincaré duality.
93
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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 ...
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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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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 \...
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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 ...
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If $M$ is compact connected non-orientable 3-manifold, then $H_1(M)$ is infinite.
Let $M$ be a compact connected non-orientable 3-manifold. The goal is to show that $H_1(M)$, the first integral homology group, is infinite.
There is a proof of this assuming $\partial M = \emptyset$. ...
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Prove/disprove that $X$ is a manifold
Let $X$ be the 2-dimensional CW complex where 2-cells are glued as follows:
The goal is to prove/disprove that $X$ is a manifold. Here's my attempt to disprove this.
Suppose $X$ is a manifold. In ...
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Correspondence between Euclidean and non-Euclidean geometries
I would like to ask correspondence between Euclidean and non-Euclidean geometries.
In the science and hypothesis, Poincare says that non-Euclidean geometry can be translated into Euclidean geometry ...
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Composition of Gysin and restriction maps on $\ell$-adic cohomology
I follow the notations of Milne's lectures notes on etale cohomology, most specifically the section titled "The Gysin map" in chapter 24, p. 145.
Let $k$ be an algebraically closed field, ...
- 5,393
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Intersection number of submanifolds with boundary
Let $N$ be an $n$-dimensional compact oriented manifold without boundary, and let $X$, $Y$ be compact oriented submanifolds of complementary dimensions $k$ and $n - k$, with boundary but such that $X \...
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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 ...
- 417
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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 ...
- 417
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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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On Poincaré Inequalities
I am trying to prove the following:
Lemma. If a metric measure space $(X,d,\mu)$ supports a $p$-Poincaré inequality for some $p \in [1;\infty)$, then it supports a $q$-Poincaré inequality for all $q \...
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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 ...
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Poincare Inequality for Mean-Zero Functions
I'm studying Sobolev spaces these days and came across a property of Poincare Inequality for mean-zero functions that I don't understand;
$u(x,y)=-u(x,-y) \implies \int_\Omega u=0$ and that Poincare ...
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Trouble with a classical fact about homology of manifold minus disk.
Let $M$ be a compact connected (smooth) $n$ dimensional manifold, and let $D$ be an $n$-dimensional disk embedded in $M$. Let $W:=\overline{M\setminus D}$, then $H_{n}(W;Z/2)=0$.
I see the following ...
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Geometric interpretation of intersection product
Let $E,E^*$ be a pair of dual $n$-dimensional real vector spaces and let $e,e^*$ be a pair of dual basis vectors of $\bigwedge^n E$ and $\bigwedge^n E^*$, respectively. For $u,v\in\bigwedge E$, we can ...
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Poincare Duality: $H^k(M) = H_{n-k}(M)^*$ or $H^k(M) = H^{n-k}(M)^*$?
There are $2$ versions of Poincare duality, in topology or in differential geometry.
Theorem(Topological Version) Let $M$ be a closed oriented $n$-manifold with fundamental class $[M]\in H_1(X)$. ...
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How to prove two partial different equations are dual
I have $m(t, x)$ and $n(t, x)$, where $t \geq 0$ is the time variable and $x \in \mathbb{R}$ is the space variable. I also have a speed $x \mapsto v(x)$. We consider the following models:
$$
\frac{\...
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Poincare duality for algebraic de Rham cohomology with integrable connection coefficients
I am reading "The Gauss-Manin Connection and Tannaka Duality" (here is the link to the paper). I am specifically interested in the proof of Proposition 2.2. In this proof, the authors use ...
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Bockstein sends fundamental class to Poincare dual of $w_1$
Suppose $X$ is an $n$-dimensional manifold, and consider the Bockstein homomorphism $\beta \colon H_k(X; \mathbb{Z}_2) \to H_{k-1}(X; \mathbb{Z})$ induced by the sequence
$$0 \to \mathbb{Z} \...
- 591
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Version of Thom isomorphism theorem for fibers with Poincaré duality
Let $F$ be a path-connected Poincaré duality space (using $\mathbb{Z}$ coefficients for singular co/homology) of dimension $n$. In particular, we have $H^n(F;\mathbb{Z}) \approx \mathbb{Z} \approx H_n(...
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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 ...
- 1,846
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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 ...
- 2,098
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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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Definition of Poincaré homomorphism
In the following question $H^i_c(M;G)$ denotes the compact support cohomology, $M$ an $n-$oriented manifold and $G$ an abelian group.
The definition I have of Poincarè homomorphism $P_M : H^i_c(M;G) \...
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Poincaré Duality Theorem (Massey proof)
I'm trying to understand the Case $1$ of Poincaré Duality Theorem (Massey p.$361$).
Theorem: Let $M$ be an $n-$oriented manifold and $G$ an arbitrary
abelian group. Then the homomorphism $$P : H_c^q(...
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A duality generalizing Lefschetz duality of a non-compact manifold
I am trying to solve out Exercise 3.3.35 of Hatcher's Book on Algebraic Topology:
I have two basic questions:
How to define the long exact sequence in the first row? For example, can we first find a ...
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Is compactly supported de Rham cohomology always finite dimensional?
I know that for a compact smooth manifold $M$, $\dim H^p_{dR}(M) < \infty$. I am trying to prove Poincaré duality, and in one of my steps so far, the compactly supported cohomology groups being ...
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Induced map on cohomology is preimage in geometry
My question is about p. 69 of Bott Tu. I don't understand how the commutative diagram implies that if $\omega$ is the cohomology class on $M$ representing $S$, then $f^*\omega$ represents $f^{-1}(S)$. ...
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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)$...
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If $X$ is an orientable connected non compact $n$- manifold then $H_n(X) = 0$
To prove the assert I led back to prove that $H_c^0(X) = 0$ thanks to Poincarè-duality, using the fact that $H_c^0(X) = \text{lim}H^0(X,X\setminus K,\mathbb{Z}) = \text{lim} 0 = 0$, since here we ...
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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-...
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Different approach of Poincaré Duality
I am studying Algebraic Topology on my own. First I would like to tell something about me. Mainly I am focused about Hyperbolic Geometry, Geometry and Topology of 3-manifolds, knot theory, etc. I am ...
user886636
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Lefschetz duality pairing on images of projections from long exact sequence
I am redoing this question as the first rendition has been a bit of a mess.
Throughout this question, we want to only consider homology with coefficients in $\mathbb{Q}$, so we can ignore issues that ...
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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)$.
...
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the difference between chains and cochains
I have heard the following definitions, for concreteness I will refer a simplicial complex $K$ and a ring $R$, though the definitions can be extended:
A chain is a formal linear combination of ...
- 1,151
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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 \...
- 3,266
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Co-H-spaces which are also Poincaré duality spaces
In this question, it is proven that any manifold $M$ which is also a Co-H space is in fact a simply-connected homology sphere. That is, $M$ is a manifold, is simply connected, and has the homology of ...
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Action on cohomology groups using Poincare duality
on the one hand it is clear, that for a degree $d$ map $f: \mathbb{P_\mathbb{C}}^1 \to \mathbb{P_\mathbb{C}}^1$ induces multiplication by $d$ on cohomology. On the other hand, if I use Poincare ...
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Homology of a compact and orientable manifold boundary
Let $X$ be a compact orientable n-manifold, $Y=\partial X$, and $R$ a ring. Suppose that $X$ is an $R$-homology ball, i.e., $H_*(X;R)\approx H_*(B^n;R)$.
The task is to compute $H_*(Y;R)$ and ...
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Definition of Poincare Duality: Integral descend to cohomology
In Bott & Tu's "Differential Forms in Algebraic Topology" on page 50 & 51, the authors define the Poincare dual of a submanifold as follows:
Let $M$ be an oriented smooth manifold of ...
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"Homotopy theoretic" form of Poincaré duality
The ncatlab page on Poincaré duality states the following:
Let $X$ be a Poincaré duality space of dimension $d$. Then there is a quasi-isomorphism
$$C^*(X) \rightarrow \Sigma^{-d}C_*(X)$$
between the ...
- 3,266
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Why can any closed manifold be constructed a fundamental class of coefficient $\mathbb{Z/2Z}$?
I am reading now a Milnor and Stasheff's "Characteristic Classes".
In the page 274, there is a sentence "The case $\Lambda= \mathbb{Z/2}$ is particularly important, since the mod 2 ...
3
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answer
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Poincare duality for reduced homology
Reduced homology
In my understanding, the reduced homology is better-behaved than the usual singular homology because the $0$th reduced homology
counts the non-trivial "closed" $0$-chain ...
- 411
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Application of the universal coefficient Theorem
I want to prove that $\dim_{\mathbb{Z}_2}(H_k(M;\mathbb{Z}_2))=\dim_{\mathbb{Z}_2}(H_{n-k}(M,\partial M;\mathbb{Z}_2))$.
This seems like a trivial result from Poincaré-Duality, so I tried
\begin{...
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commutative square up to sign, Poincaré duality
Let $M$ be smooth oriented manifold with boundary $\partial M $, dim$M=n$. The two short exact sequences in de Rham cohomology and singular homology
$$0\longrightarrow{}\Omega^{}(M, \partial M)\...
