Jonginn Yun

Homology

Homology assigns algebraic data to holes of different dimensions. Very roughly, one studies chains, cycles, and boundaries:

$$\partial_{k}\partial_{k+1}=0.$$

Cycles with no boundary may still be boundaries of higher-dimensional chains. Homology classes are cycles modulo boundaries.

Cohomology

Cohomology is the dual theory. Instead of chains, it uses cochains, and the coboundary operator satisfies

$$d^2=0.$$

In de Rham cohomology, cochains are differential forms. Closed forms satisfy $d\omega=0$, and exact forms have the form $\omega=d\eta$.

The quotient

$$H^k_{\mathrm{dR}}(M)=\frac{\ker d:\Omega^k(M)\to\Omega^{k+1}(M)}{\operatorname{im}d:\Omega^{k-1}(M)\to\Omega^k(M)}$$

captures global information invisible to local calculus.

Characteristic Classes

Characteristic classes assign cohomology classes to vector bundles. They measure global twisting that cannot be removed by local trivialization.

For a complex line bundle with curvature two-form $F$, the first Chern class is represented by

$$\frac{F}{2\pi}$$

up to conventional factors depending on whether the connection is written as real or imaginary-valued.

On a closed oriented surface, the first Chern number is

$$\frac{1}{2\pi}\int F.$$

This is the algebraic-topology backbone behind Berry curvature, Bloch bundles, and Chern insulators.