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 preschool child does not emerge from your toddler on a given date or birthday. He becomes a child when he ceases to be a wayward, confusing, unpredictable and often balky person-in-the- making, and becomes a comparatively cooperative, eager-and-easy-to-please real human being—at least 60 per cent of the time.”
    —Penelope Leach (20th century)

    “In the tale proper—where there is no space for development of character or for great profusion and variety of incident—mere construction is, of course, far more imperatively demanded than in the novel.”
    —Edgar Allan Poe (1809–1849)