Fundamental class

A fundamental class is a homology class of an oriented variety that corresponds to the “whole variety”. An intuitively fundamental class can be imagined as the sum of simplices of the maximum dimension of a suitable triangulation of a manifold.

Fundamental manifold class $M$ ${\ displaystyle M}$ $M$ usually denoted by $[M]$ ${\ displaystyle [M]}$ ${\ displaystyle [M]}$ .

Content

Definition

Closed orientable manifold

If the variety $M$ ${\ displaystyle M}$ dimensions $n$ ${\ displaystyle n}$ is connected orientable and closed , then $n$ ${\ displaystyle n}$ -th homology group is infinite cyclic : $H_{n}(M,\mathbf {Z} )\cong \mathbf {Z}$ ${\ displaystyle H_ {n} (M, \ mathbf {Z}) \ cong \ mathbf {Z}}$ . The orientation of the manifold is determined by the choice of the generating element of the group or isomorphism $\mathbf {Z} \to H_{n}(M,\mathbf {Z} )$ ${\ displaystyle \ mathbf {Z} \ to H_ {n} (M, \ mathbf {Z})}$ . The generating element is called the fundamental class .

Formally incoherent orientable manifold $M=\cup _{i}M_{i}$ ${\ displaystyle M = \ cup _ {i} M_ {i}}$ as a fundamental class you can compare the amount $\sum [M_{i}]$ ${\ displaystyle \ sum [M_ {i}]}$ fundamental classes of all its connected components $M_{i}$ ${\ displaystyle M_ {i}}$ . However, this element is not a group spawner. $H_{n}(M,\mathbf {Z} )=\oplus H_{n}(M_{i},\mathbf {Z} )=\mathbf {Z} \oplus \dots \oplus \mathbf {Z}$ ${\ displaystyle H_ {n} (M, \ mathbf {Z}) = \ oplus H_ {n} (M_ {i}, \ mathbf {Z}) = \ mathbf {Z} \ oplus \ dots \ oplus \ mathbf { Z}}$ .

Non-orientable manifold

For a non-orientable manifold, the group $H_{n}(M;\mathbf {Z} )=0$ ${\ displaystyle H_ {n} (M; \ mathbf {Z}) = 0}$ if in this case M is connected and closed, then $H_{n}(M;\mathbf {Z} _{2})=\mathbf {Z} _{2}$ ${\ displaystyle H_ {n} (M; \ mathbf {Z} _ {2}) = \ mathbf {Z} _ {2}}$ . The parent element of the group $H_{n}(M;\mathbf {Z} _{2})$ ${\ displaystyle H_ {n} (M; \ mathbf {Z} _ {2})}$ is called the fundamental class of a non-orientable manifold M.

$\mathbf {Z} _{2}$ ${\ displaystyle \ mathbf {Z} _ {2}}$ -fundamental class of a variety is used in determining the Stiefel-Whitney numbers .

Edge Variety

If M is a compact orientable manifold with boundary $\partial M$ ${\ displaystyle \ partial M}$ , the n -th relative homology group is infinite cyclic : $H_{n}(M,\partial M)\cong \mathbf {Z}$ ${\ displaystyle H_ {n} (M, \ partial M) \ cong \ mathbf {Z}}$ . The generating element of the group $H_{n}(M,\partial M)$ ${\ displaystyle H_ {n} (M, \ partial M)}$ is called the fundamental class of a manifold with a boundary.

Poincare duality

The main result of the homological theory of varieties is Poincaré duality between the homology groups and the cohomology of the manifold. Corresponding Poincare isomorphism $D:H^{k}(M;\mathbf {Z} )\to H_{n-k}(M;\mathbf {Z} )$ ${\ displaystyle D: H ^ {k} (M; \ mathbf {Z}) \ to H_ {nk} (M; \ mathbf {Z})}$ (for oriented) and $D:H^{k}(M;\mathbf {Z} _{2})\to H_{n-k}(M;\mathbf {Z} _{2})$ ${\ displaystyle D: H ^ {k} (M; \ mathbf {Z} _ {2}) \ to H_ {nk} (M; \ mathbf {Z} _ {2})}$ (for a non-orientable) manifold is determined by the corresponding fundamental class of the manifold:

D(\alpha )=[M]\frown \alpha

{\ displaystyle D (\ alpha) = [M] \ frown \ alpha}

,

Where $\frown$ ${\ displaystyle \ frown}$ denotes $\frown$ ${\ displaystyle \ frown}$ -the multiplication of homological and cohomological classes.

Degree of display

If a $M$ ${\ displaystyle M}$ , $N$ ${\ displaystyle N}$ - connected closed oriented manifolds of the same dimension, and $f:M\to N$ ${\ displaystyle f: M \ to N}$ - continuous mapping , then

f_{*}[M]=k[N]

{\ displaystyle f _ {*} [M] = k [N]} ,

Where $k$ ${\ displaystyle k}$ - some integer . This number is called the degree of display. $f$ ${\ displaystyle f}$ and is denoted by deg f .

Literature

A.T. Fomenko, D.B. Fuchs . The course of homotopic topology - M: Nauka, 1989.
A. Dold Lectures on algebraic topology - M: Mir, 1976.