Universal Coefficient Theorem - Example: mod 2 Cohomology of The Real Projective Space

Example: mod 2 Cohomology of The Real Projective Space

Let X = Pn(R), the real projective space. We compute the singular cohomology of X with coefficients in R := Z2.

Knowing that the integer homology is given by:

H_i(X; \mathbf{Z}) = \begin{cases} \mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\ \mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\ 0 & \mbox{else.} \end{cases}

We have Ext(R, R) = R, Ext(Z, R)= 0, so that the above exact sequences yield

.

In fact the total cohomology ring structure is

.

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