Let U,V be open cover of circle - S1. Let X,Y be arcs on the circle that correspond to the open cover with disjoint overlaps at top of the circle and at the bottom of the circle.
Each arc X,Y is diffeomorphic to an interval and thus to R.
Here instead of writing one forms and seeking existence of integral solutions, we can use Mayer-Vietoris sequence to make computations. The Mayer-Vietoris sequence for S1 is as follows:
0→H0(M)i∗→H0(U)⊕H0(V)j∗→H0(U∩V)d∗→H1(M)→0.
Using dimensional formula for sequence of vector spaces ∑nk=0(−1)kdk, we can figure out dimension of H1(M)=d1=1 as follows:
1−2+2−d1=0
Since S1 is connected, H0(S1)=R. As shown before H0(U)=H0(V)=R. Since overlaps are disjoint we have R⊕R. All this results in the following sequence.
0→Ri∗→R⊕Rj∗→R⊕R→0
Notice j∗:H0(U)⊕H0(V)j∗→H0(U∩V) is given as follows. Since, we are dealing with 0 dimensional space, corresponding vectors are 0 dimensionals - that is scalars or real numbers.
j∗(m,n)=(n−m,n−m)
That is these are elements of diagonal in R×R. d∗:H0(U∩V)d∗→H1(M) sends this to one dimensional space which is isomorphic to R, hence points m≠n will result in this element. Among many, we can choose one.
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