Geodesic
Manifolds of the homotopy type of the product of two projective spaces
Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 471-486

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In this paper the sets of homotopy triangulations of the manifold $RP^k\times RP^l$ are computed for all $k,l\geqslant3$ except for the case when $k$ and $l$ are even and $k+l=4r$. Bibliography: 11 titles. 
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     author = {A. F. Kharshiladze},
     title = {Manifolds of the homotopy type of the product of two projective spaces},
     journal = {Sbornik. Mathematics},
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     year = {1975},
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A. F. Kharshiladze. Manifolds of the homotopy type of the product of two projective spaces. Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 471-486. http://geodesic.mathdoc.fr/item/SM_1975_25_4_a0/