Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 114 / 1 / i

Solution

RP^{n}

has one cell in each dimension

0, 1, \dots, n

C_{q} ≅ Z

and differential

d_{q} = 1 + (- 1)^{q} = {2, 0, q even, q odd .

Consequently

H_{q} (RP^{n}; Z) ≅ ⎩ ⎨ ⎧ Z, Z /2, Z, 0, q = 0, 0 < q < n and q odd, q = n and n odd, otherwise .

After tensoring the cellular complex with

F_{2} = Z /2

, every differential vanishes, so

H_{q} (RP^{n}; F_{2}) ≅ {F_{2}, 0, 0 \leq q \leq n, q > n .

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