Manifolds and duality · Qual Corpus

For an $n$-manifold $M$ and $x\in M$, $H_n(M,M\setminus\{x\};\ZZ)\cong\ZZ$. An orientation of $M$ is a choice of generator $\mu_x$ of this group for every $x$ that is locally consistent: each $x$ has a ball neighbourhood $B$ and a class in $H_n(M,M\setminus B;\ZZ)$ restricting to $\mu_y$ for every $y\in B$. The orientation double cover $\tilde M\to M$ has fibre over $x$ the two generators of $H_n(M,M\setminus\{x\};\ZZ)$, and a connected $M$ is orientable if and only if $\tilde M$ is disconnected. A closed connected orientable $n$-manifold has a fundamental class $[M]$ generating $H_n(M;\ZZ)\cong\ZZ$, and Poincaré duality states that $H^k(M;\ZZ)\to H_{n-k}(M;\ZZ)$, $\varphi\mapsto[M]\frown\varphi$, is an isomorphism for every $k$. For a compact orientable $n$-manifold $M$ with boundary, Lefschetz duality gives $H^k(M;\ZZ)\cong H_{n-k}(M,\partial M;\ZZ)$ and $H^k(M,\partial M;\ZZ)\cong H_{n-k}(M;\ZZ)$. For a compact, locally contractible, nonempty proper subspace $K\subseteq S^n$, Alexander duality gives $\tilde H_i(S^n\setminus K;\ZZ)\cong\tilde H^{n-i-1}(K;\ZZ)$. For a closed oriented $2m$-manifold, $(\alpha,\beta)\mapsto\langle\alpha\smile\beta,[M]\rangle$ on $H^m(M;\ZZ)$ modulo torsion is the intersection form, a unimodular bilinear form that is symmetric for $m$ even and skew-symmetric for $m$ odd.

Manifold (Tag D-UBWVX)

Let $n\geq0$. An $n$-manifold is a Hausdorff space $M$ in which every point has an open neighborhood homeomorphic to $\RR^n$ [@Hat02].

The definition of a topological manifold in [@Lee13] also requires $M$ to be second-countable. The open long line is a Hausdorff space in which every point has an open neighborhood homeomorphic to $\RR$, and it is not second-countable.

Manifold with boundary (Tag D-MN6QW)

Let $n\geq 1$ and let $\mathbb H^n\coloneqq\ts{x\in\RR^n \st x_n\geq 0}$ be the closed upper half-space, with the subspace topology. A topological $n$-manifold with boundary is a Hausdorff space $M$ in which every point has an open neighborhood homeomorphic to $\RR^n$ or to $\mathbb H^n$.

Closed manifold (Tag D-SYKQW)

A closed manifold is a compact manifold without boundary [@Hat02].

Dimension of a manifold (Tag D-LOISU)

Let $M$ be an $n$-manifold and $x\in M$. Then $H_i(M, M\sm\ts{x};\ZZ)\cong\ZZ$ for $i = n$ and $H_i(M, M\sm\ts{x};\ZZ) = 0$ for $i\neq n$. In particular, a nonempty space that is both an $m$-manifold and an $n$-manifold has $m = n$.

Let $M$ be a nonempty space that is an $n$-manifold for some $n\geq 0$. The dimension of $M$ is this integer $n$, equivalently the unique $i$ with $H_i(M, M\sm\ts{x};\ZZ)\neq 0$ for any $x\in M$.

See [@Hat02].

Local orientation of a manifold (Tag D-JW63I)

Let $M$ be an $n$-manifold and $x\in M$. Then $H_n(M, M\sm\ts{x};\ZZ)\cong H_n(\RR^n, \RR^n\sm\ts{0};\ZZ)\cong\ZZ$.

Let $M$ be an $n$-manifold and $x\in M$. A local orientation of $M$ at $x$ is a choice of generator $\mu_x$ of the infinite cyclic group $H_n(M, M\sm\ts{x};\ZZ)$.

See [@Hat02].

Oriented manifold (Tag D-CNLBT)

Let $M$ be an $n$-manifold. An orientation of $M$ is a function $x\mapsto\mu_x$ assigning to each $x\in M$ a local orientation $\mu_x$, a generator of $H_n(M, M\sm\ts{x};\ZZ)\cong\ZZ$, such that every $x\in M$ has an open ball neighborhood $B$ of finite radius in a chart $\RR^n\subseteq M$ and a class $\mu_B\in H_n(M, M\sm B;\ZZ)$ whose image under the map $H_n(M, M\sm B;\ZZ)\to H_n(M, M\sm\ts{y};\ZZ)$ induced by inclusion is $\mu_y$ for every $y\in B$. An oriented manifold is a pair consisting of a manifold and an orientation of it. The manifold $M$ is orientable if an orientation of $M$ exists.

A nonempty connected orientable $n$-manifold has exactly two orientations.

Let $M$ be a closed connected oriented $n$-manifold with orientation $\mu$. There is a unique class $[M]\in H_n(M;\ZZ)$ whose image in $H_n(M, M\sm\ts{x};\ZZ)$ is $\mu_x$ for every $x\in M$, and $[M]$ generates $H_n(M;\ZZ)\cong\ZZ$.

The class $[M]$ is the fundamental class of $M$.

See [@Hat02].

Local orientation and orientation of a manifold (Tag D-6CI7D)

Let $M$ be an $n$-manifold, and for $A\subseteq M$ write $H_n(M\mid A)\coloneqq H_n(M, M\sm A;\ZZ)$. For $x\in M$, excision gives $H_n(M\mid x)\cong H_n(\RR^n, \RR^n\sm\ts{0};\ZZ)\cong\ZZ$.

A local orientation of $M$ at $x$ is a generator $\mu_x$ of $H_n(M\mid x)$.
An orientation of $M$ is a function $x\mapsto\mu_x$ assigning to each $x\in M$ a local orientation at $x$, such that each $x\in M$ has an open neighborhood $U\subseteq M$ with a homeomorphism $U\cong\RR^n$ and an open ball $B\subseteq U$ of finite radius containing $x$, together with a generator $\mu_B$ of $H_n(M\mid B)\cong\ZZ$ whose image under the map $H_n(M\mid B)\to H_n(M\mid y)$ induced by inclusion is $\mu_y$ for every $y\in B$.
$M$ is orientable if an orientation of $M$ exists.

Each $H_n(M\mid x)$ has exactly two generators, so $M$ has exactly two local orientations at each point. The local orientations form the orientation double cover $\tilde M = \ts{\mu_x \st x\in M,\ \mu_x \text{ a local orientation of } M \text{ at } x}$, with $\mu_x\mapsto x$. The manifold $\tilde M$ is orientable, and a connected manifold $M$ is orientable if and only if $\tilde M$ has two components [@Hat02].

Let $M$ be a connected $n$-manifold with orientation double cover $p\colon\tilde M\to M$, and let $x\in M$. The following are equivalent:

$M$ is orientable.
$\tilde M$ is disconnected.
$\tilde M$ has two components, each mapped homeomorphically onto $M$ by $p$.
$p$ has a continuous section $s\colon M\to\tilde M$, $p\circ s = \id_M$.
The homomorphism $\pi_1(M, x)\to\ZZ/2$ given by the action of $\pi_1(M,x)$ on the two-point fiber $\inverseof{p}(x)$ is trivial.

[@Hat02].

$R$-orientability (Tag D-YD6DR)

Let $R$ be a commutative ring with identity and $M$ an $n$-manifold, and for $A\subseteq M$ write $H_n(M\mid A;R)\coloneqq H_n(M, M\sm A; R)$, so that $H_n(M\mid x;R)\cong R$ for $x\in M$. An $R$-orientation of $M$ is a function $x\mapsto\mu_x$ assigning to each $x\in M$ a generator $\mu_x$ of the $R$-module $H_n(M\mid x;R)$ that is locally consistent: each $x\in M$ has a neighborhood $\RR^n\subseteq M$ containing an open ball $B$ of finite radius about $x$ and an element $\mu_B\in H_n(M\mid B;R)$ whose image under the map $H_n(M\mid B;R)\to H_n(M\mid y;R)$ induced by inclusion is $\mu_y$ for every $y\in B$. The manifold $M$ is $R$-orientable if it has an $R$-orientation.

Let $M$ be an $n$-manifold.

$M$ is $\ZZ/2$-orientable.
If $M$ is orientable, then $M$ is $R$-orientable for every commutative ring $R$ with identity.
If $R$ is a commutative ring with identity in which $2\neq 0$, then $M$ is $R$-orientable if and only if $M$ is orientable.

[@Hat02].

Orientation cover (Tag D-WX7JH)

Let $M$ be an $n$-manifold, and for $A\subseteq M$ write $H_n(M\mid A) \coloneqq H_n(M, M\sm A;\ZZ)$. The orientation double cover of $M$ is the set $\tilde M$ of all local orientations $\mu_x$ of $M$ at points $x\in M$, with the map $p\colon\tilde M\to M$, $\mu_x\mapsto x$, and the topology with basis the sets \[ U(\mu_B) \coloneqq \ts{ \mu_x \st x\in B,\ \mu_x \text{ the image of } \mu_B \text{ under } H_n(M\mid B)\to H_n(M\mid x) }, \] where $B$ ranges over the open balls of finite radius in charts $\RR^n\subseteq M$ and $\mu_B$ over the generators of $H_n(M\mid B)\cong\ZZ$.

Let $M$ be an $n$-manifold with orientation double cover $p\colon\tilde M\to M$. Then $p$ is a two-sheeted covering space, and $\tilde M$ is an orientable $n$-manifold. If $M$ is connected, then $M$ is orientable if and only if $\tilde M$ has two components.

[@Hat02].

Orientation double cover (Tag PR-BQKHS)

Every manifold has an orientable double cover; it disconnects exactly when the base is orientable.Suggest a slogan

Every $n$-manifold $M$ has an orientable two-sheeted covering space $\tilde M\to M$, the orientation double cover [@Hat02]. If $M$ is connected, then $M$ is orientable if and only if $\tilde M$ has two components; in particular, $M$ is orientable if $\pi_1(M)$ has no subgroup of index two [@Hat02].

Fundamental class (Tag D-TS7TZ)

Let $M$ be a closed connected $n$-manifold with an orientation $x\mapsto\mu_x\in H_n(M,M\setminus\ts{x};\ZZ)$. A fundamental class of $M$ is a class $[M]\in H_n(M;\ZZ)$ whose image under $H_n(M;\ZZ)\to H_n(M,M\setminus\ts{x};\ZZ)$ is $\mu_x$ for every $x\in M$.

Let $M$ be a closed connected $n$-manifold with an orientation $x\mapsto\mu_x$. Then $H_n(M;\ZZ)\cong\ZZ$, the fundamental class $[M]$ exists and is unique, and it generates $H_n(M;\ZZ)$ [@Hat02].

Top homology of closed manifolds (Tag PR-LR35S)

A closed connected manifold has top integral homology $\ZZ$ exactly when it is orientable.Suggest a slogan

Let $M$ be a closed connected $n$-manifold. Then \[ H_n(M;\ZZ) \cong \begin{cases} \ZZ & M \text{ orientable},\\ 0 & \text{otherwise}, \end{cases} \qquad H_i(M;\ZZ) = 0 \text{ for } i > n \] [@Hat02], and the torsion subgroup of $H_{n-1}(M;\ZZ)$ is trivial if $M$ is orientable and $\ZZ/2$ otherwise [@Hat02].

Vanishing of homology and cohomology above the dimension (Tag PR-ZCPDD)

A closed $n$-manifold has no homology or integral cohomology above degree $n$.Suggest a slogan

Let $M$ be a closed connected $n$-manifold. Then $H_i(M;R) = 0$ for $i > n$ and every coefficient ring $R$ [@Hat02], and $H^i(M;\ZZ) = 0$ for $i>n$ by the universal coefficient theorem [@Hat02], since $H_n(M;\ZZ)$ is free.

In degree $n$, $H^n(M;\ZZ)\cong\ZZ$ if $M$ is orientable and $H^n(M;\ZZ)\cong\ZZ/2$ otherwise, by the universal coefficient theorem applied to $H_n(M;\ZZ)$ and the torsion subgroup of $H_{n-1}(M;\ZZ)$ [@Hat02].

Cup product (Tag D-B2JER)

Let $X$ be a topological space, $R$ a ring, and $p, q\geq 0$. For singular cochains $\varphi\in C^p(X;R)$ and $\psi\in C^q(X;R)$, the cup product $\varphi\smile\psi\in C^{p+q}(X;R)$ is the cochain whose value on a singular $(p+q)$-simplex $\sigma\colon[v_0, \ldots, v_{p+q}]\to X$ is \[ (\varphi\smile\psi)(\sigma) \coloneqq \varphi\qty{\ro{\sigma}{[v_0, \ldots, v_p]}}\,\psi\qty{\ro{\sigma}{[v_p, \ldots, v_{p+q}]}} ,\] where each face is identified with the standard simplex of its dimension by the order-preserving linear homeomorphism.

The coboundary satisfies $\delta(\varphi\smile\psi) = \delta\varphi\smile\psi + (-1)^p\varphi\smile\delta\psi$, so the cup product of two cocycles is a cocycle and the cup product of a cocycle with a coboundary is a coboundary. Hence it induces a bilinear map \[ \smile\colon H^p(X;R)\cross H^q(X;R)\to H^{p+q}(X;R), \qquad [\varphi]\smile[\psi]\coloneqq[\varphi\smile\psi] .\]

[@Hat02].

Cap product (Tag D-RQS4J)

Let $X$ be a topological space, $R$ a commutative ring, and $0\leq q\leq p$. For a singular $p$-simplex $\sigma\colon\Delta^p=[v_0,\ldots,v_p]\to X$ and $0\leq i\leq j\leq p$, write $\sigma|_{[v_i,\ldots,v_j]}$ for the singular $(j-i)$-simplex obtained by restricting $\sigma$ to the face $[v_i,\ldots,v_j]$, identified with $\Delta^{j-i}$ preserving the order of the vertices. The cap product on chains and cochains is the $R$-bilinear map \[ \frown\colon C_p(X;R)\times C^q(X;R)\to C_{p-q}(X;R),\qquad \sigma\frown\psi\coloneqq\psi\qty{\sigma|_{[v_0,\ldots,v_q]}}\,\sigma|_{[v_q,\ldots,v_p]} \] on singular $p$-simplices $\sigma$ and singular $q$-cochains $\psi$, extended linearly in $\sigma$ [@Hat02].

For $\sigma\in C_p(X;R)$ and $\psi\in C^q(X;R)$, \[ \del(\sigma\frown\psi)=(-1)^q\qty{\del\sigma\frown\psi-\sigma\frown\delta\psi}, \] so the cap product of a cycle and a cocycle is a cycle, and it induces an $R$-bilinear map \[ \frown\colon H_p(X;R)\times H^q(X;R)\to H_{p-q}(X;R) \] [@Hat02].

Kronecker pairing (Tag D-VP4LC)

Let $X$ be a topological space, $R$ a commutative ring, and $n\geq 0$. For a singular cochain $\varphi\in C^n(X;R) = \Hom_R(C_n(X;R), R)$ and a singular chain $\alpha\in C_n(X;R)$, put $\inner{\varphi}{\alpha} \coloneqq \varphi(\alpha)$. The Kronecker pairing, also called the Kronecker product, is the $R$-bilinear map \[ \inner{\wait}{\wait}\colon H^n(X; R) \cross H_n(X; R) \to R, \qquad \inner{[\varphi]}{[\alpha]} \coloneqq \varphi(\alpha), \] for a cocycle $\varphi$ and a cycle $\alpha$. Its adjoint is the evaluation homomorphism \[ h\colon H^n(X; R) \to \Hom_R(H_n(X;R), R), \qquad h([\varphi])([\alpha]) \coloneqq \varphi(\alpha). \]

Let $X$ be a topological space, $R$ a commutative ring, and $n\geq 0$.

For $\psi\in C^{n-1}(X;R)$ and $\alpha\in C_n(X;R)$, $\inner{\delta\psi}{\alpha} = \inner{\psi}{\del\alpha}$; hence the Kronecker pairing is well defined on classes.
For a continuous map $f\colon X\to Y$, $\beta\in H^n(Y;R)$, and $x \in H_n(X;R)$, $\inner{f^*\beta}{x} = \inner{\beta}{f_*x}$.
If $R$ is a principal ideal domain, there is a split short exact sequence \[ 0 \to \Ext_R(H_{n-1}(X;R), R) \to H^n(X;R) \xrightarrow{h} \Hom_R(H_n(X;R), R) \to 0, \] so $h$ is surjective, and $h$ is an isomorphism if and only if $\Ext_R(H_{n-1}(X;R), R) = 0$.

The evaluation homomorphism $h$ need not be injective. For $X = \RR P^2$, $R = \ZZ$, and $n = 2$, one has $H_1(\RR P^2;\ZZ)\cong\ZZ/2$ and $H_2(\RR P^2;\ZZ) = 0$, so $H^2(\RR P^2;\ZZ)\cong\Ext_\ZZ(\ZZ/2,\ZZ)\cong\ZZ/2$ while $\Hom_\ZZ(H_2(\RR P^2;\ZZ),\ZZ) = 0$.

[@Hat02].

Poincaré duality (Tag D-QP7WI)

Let $R$ be a commutative ring and $M$ a closed $R$-orientable $n$-manifold with fundamental class $[M]\in H_n(M;R)$. The Poincaré duality map is \[ D\colon H^k(M;R)\to H_{n-k}(M;R),\qquad D(\alpha)\coloneqq[M]\frown\alpha, \] given by the cap product with $[M]$.

Let $R$ be a commutative ring and $M$ a closed $R$-orientable $n$-manifold with fundamental class $[M]\in H_n(M;R)$. Then the Poincaré duality map $D\colon H^k(M;R)\to H_{n-k}(M;R)$, $\alpha\mapsto[M]\frown\alpha$, is an isomorphism for every $k$ [@Hat02].

Poincaré duality for closed manifolds (Tag PR-4X6G2)

On a closed orientable manifold, cap product with the fundamental class exchanges degree $k$ cohomology with degree $n-k$ homology.Suggest a slogan

Let $M$ be a closed orientable $n$-manifold and $R$ a commutative ring; for example $R = \ZZ$ or $R$ a field. Then cap product with the fundamental class gives isomorphisms \[ H^{k}(M; R) \xrightarrow{\ \sim\ } H_{n-k}(M; R) \] for all $k$ [@Hat02].

For $n\geq 1$, the orientable manifold $\RR^n$ is not closed, and $H^0(\RR^n;\ZZ)\cong\ZZ$ while $H_n(\RR^n;\ZZ) = 0$. The closed manifold $\RP^2$ is not orientable, and $H^0(\RP^2;\ZZ)\cong\ZZ$ while $H_2(\RP^2;\ZZ) = 0$. For a field $F$ of characteristic $2$, every closed manifold is $F$-orientable, so the duality holds for all closed manifolds with $\ZZ/2$ coefficients.

Lefschetz duality (Tag T-QNYSB)

For manifolds with boundary, cap product exchanges relative cohomology on one boundary piece with homology relative to the other.Suggest a slogan

Let $M$ be a compact $R\dash$orientable $n\dash$manifold whose boundary is decomposed as $\del M = A \union B$ with $A, B$ compact $(n-1)\dash$manifolds meeting in $\del A = \del B = A \intersect B$. Then cap product with a fundamental class $[M]\in H_n(M, \del M; R)$ gives isomorphisms \[ D_M: H^k(M, A; R) \mapsvia{\sim} H_{n-k}(M, B; R) \] for all $k$. The cases $A = \del M$, $B = \emptyset$ and $A = \emptyset$, $B = \del M$ give the isomorphisms \[ H^k(M, \del M; R) \cong H_{n-k}(M; R), \qquad H^k(M; R)\cong H_{n-k}(M, \del M; R) \] [@Hat02].

For $\del M = \emptyset$ this is Poincaré duality.

Alexander duality (Tag D-5KDNB)

The reduced homology of a complement in $S^n$ is the shifted reduced cohomology of the subset.Suggest a slogan

Let $n\geq 1$ and let $K\subsetneq S^n$ be a nonempty, compact, locally contractible subspace. Then for all $i$, \[ \tilde H_i(S^n \sm K; \ZZ) \cong \tilde H^{n-i-1}(K;\ZZ) .\]

It follows, with $M = S^n$ and the long exact sequence of the pair $(S^n, S^n\sm K)$, from the duality $H_i(M, M\sm K;\ZZ)\cong H^{n-i}(K;\ZZ)$ for a compact, locally contractible subspace $K$ of a closed, orientable $n$-manifold $M$ [@Hat02].

[@Hat02].

Intersection form of a manifold (Tag D-6POU4)

Let $M$ be a closed, connected, oriented $2k$-manifold with fundamental class $[M]\in H_{2k}(M;\ZZ)$. The intersection form of $M$ is the bilinear form \[ I\colon H^k(M;\ZZ)\cross H^k(M;\ZZ) \to \ZZ, \qquad I(a, b)\coloneqq \inner{a\smile b}{[M]} ,\] where $\smile$ is the cup product and $\inner{\wait}{\wait}$ is the Kronecker pairing.

Graded commutativity of the cup product gives $I(a, b) = (-1)^k I(b, a)$, so $I$ is symmetric when $k$ is even, that is, when $\dim M\equiv 0 \pmod 4$, and skew-symmetric when $k$ is odd. By Poincaré duality, $I$ vanishes on torsion classes and induces a nondegenerate form on $H^k(M;\ZZ)/\text{torsion}$ whose Gram matrix has determinant $\pm1$.

[@Hat02].

Nonsingular cup product pairing (Tag PR-3FB24)

Poincaré duality makes complementary-degree cup products nonsingular after removing torsion.Suggest a slogan

Let $M$ be a closed orientable $n$-manifold. The cup product pairing \[ H^k(M;\ZZ)/\mathrm{torsion}\times H^{n-k}(M;\ZZ)/\mathrm{torsion}\to\ZZ,\qquad (a,b)\mapsto\langle a\smile b, [M]\rangle \] is nonsingular, and for a field $F$ the pairing $H^k(M;F)\times H^{n-k}(M;F)\to F$ is nonsingular for every closed $F$-orientable $n$-manifold $M$ [@Hat02]. Under Poincaré duality this is the intersection pairing on $H_{n-k}(M)\times H_k(M)$ modulo torsion.

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