smooth structure, the existence of a Morse function would show RPn is a CW complex. One such function is given by, in homogeneous coordinates, On each neighborhood Ui, g has nondegenerate critical point (0,...,1,...,0) where 1 occurs in the i-th position with Morse index i. This shows RPn is a CW complex with 1 cell in every dimension. Tautological bundles Real projective space has a natural line bundle over it, called the tautological bundle. More precisely, this is called the tautological subbundle, and there is also a dual n-dimensional bundle called the tautological quotient bundle. Algebraic topology of real projective spaces Homotopy groups The higher homotopy groups of RPn are exactly the higher homotopy groups of Sn, via the long exact sequence on homotopy associated to a fibration. Explicitly, the fiber bundle is: You might also write this as or by analogy with complex projective space. The homotopy groups are: Homology The cellular chain complex associated to the above CW structure has 1 cell in each dimension 0, ..., n. For each dimensional k, the boundary maps dk : δDk → RPk−1/RPk−2 is the map that collapses the equator on Sk−1 and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): Thus the integral homology is RPn is orientable if and only if n is odd, as the above homology calculation shows. Infinite real projective space The infinite real projective space is constructed as the direct limit or union of the finite projective spaces: This space is classifying space of O(1), the first orthogonal group. The double cover of this space is the infinite sphere , which is contractible. The infinite projective space is therefore the Eilenberg–MacLane space K(Z2, 1). For each nonnegative integer q, the modulo 2 homology group . Its cohomology ring modulo 2 is where is the first Stiefel–Whitney class: it is the free -algebra on , which has degree 1. See also Complex projective space Quaternionic projective space Lens space Real projective plane Notes References Bredon, Glen. Topology and geometry, Graduate Texts in Mathematics, Springer Verlag 1993, 1996 Algebraic topology Differential geometry Projective geometry