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:

    “A man’s real possession is his memory. In nothing else is he rich, in nothing else is he poor.”
    —Alexander Smith (1830–1867)

    “In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
    —Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)