Processing math: 100%

Monday, July 13, 2020

Cohomology of Real line.

First example of applications of Cohomology is real line R.

To start off a fact about differential forms:

Differential forms belong to spaces of Alternating forms - Ak(M). Whenever k>n where n dimension of tangent space at a give point, differential k forms become 0.

Since R is connected, can conclude, H0(R)=R. Clearly, all two forms are zero as n=1. Note, two forms are generated by one forms. Since all two forms are zero, all one forms are closed.

Note a function such h(x) is a zero form. A one form f(x)dx on R is exact if and only if there exists a C function g(x) on R such that the following is satisfied.

f(x)dx = dg = g^'(x) dx

which means,

g(x)=x0f(t)dt

Thus,

Hk(R)=R when k=0, and Hk(R)=0 when k>0.




No comments:

Post a Comment

Chain complexes on Hilbert spaces

 Chain complexes are mathematical structures used extensively in algebraic topology, homological algebra, and other areas of mathematics. Th...