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:

    “O.J. Berman: Well, answer the question now. Is she or isn’t she?
    Paul: Is she or isn’t she what?
    O.J. Berman: A phony.
    Paul. I don’t know. I don’t think so.
    O.J. Berman: You don’t think so, huh? Well, you’re wrong. She is. But on the other hand, you’re right. Because she’s a real phony.”
    —George Axelrod (b. 1922)

    “Not so many years ago there there was no simpler or more intelligible notion than that of going on a journey. Travel—movement through space—provided the universal metaphor for change.... One of the subtle confusions—perhaps one of the secret terrors—of modern life is that we have lost this refuge. No longer do we move through space as we once did.”
    —Daniel J. Boorstin (b. 1914)