Rp×n

Rp×n denotes the Cartesian product RP^p × RP^n, where RP^k is the real projective k-space. RP^k is the space of lines through the origin in R^{k+1}, equivalently the quotient S^k/±1. It is a compact, connected manifold of dimension k for p,n ≥ 1 and has a standard CW structure with one cell in each dimension up to k.

As a product, Rp×n inherits its topology from the factors and has dimension p+n. Its basic properties arise from those of RP^p and RP^n, and it is frequently used as a test case in algebraic topology and bundle theory.

The fundamental group of Rp×n is the product of the fundamental groups: π1(RP^p × RP^n) ≅ π1(RP^p) × π1(RP^n). For p,n ≥ 2, each factor has π1 ≅ Z/2, so π1(RP^p × RP^n) ≅ Z/2 × Z/2. If one factor is RP^1 ≅ S^1, the product may also include a Z factor accordingly. The space is orientable only when both p and n are odd; in general RP^k is orientable iff k is odd, and a product is orientable exactly when both factors are.

Cohomology with Z/2 coefficients is straightforward: H^*(RP^p × RP^n; Z/2) ≅ Z/2[a,b]/(a^{p+1}, b^{n+1}) with deg(a)=deg(b)=1. Integer cohomology is more intricate due to torsion, but the Kunneth theorem relates H^*(RP^p × RP^n; Z) to that of the factors.

Rp×n serves as a fundamental example in studying product manifolds, characteristic classes, and the interaction of cohomology rings under Cartesian products.