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.
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