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

.

Read more about this topic:  Universal Coefficient Theorem

Famous quotes containing the words real and/or space:

    The real community of man ... is the community of those who seek the truth, of the potential knowers.
    Allan Bloom (1930–1992)

    The secret ones around a stone
    Their lips withdrawn in meet surprise
    Lie still, being naught but bone
    With naught but space within their eyes....
    Allen Tate (1899–1979)