Geodesic
Involution of manifolds with a set of fixed points diffeomorphic to real projective space
Matematičeskie zametki, Tome 9 (1971) no. 3, pp. 249-252

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It is proved that if, on a manifold with an involution, the subset of fixed points is diffeomorphic to an even-dimensional real projective space, then the manifold is bordant to the complex projective space in the class of nonoriented bordisms. 
@article{MZM_1971_9_3_a2,
     author = {A. S. Mishchenko},
     title = {Involution of manifolds with a set of fixed points diffeomorphic to real projective space},
     journal = {Matemati\v{c}eskie zametki},
     pages = {249--252},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_3_a2/}
}
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A. S. Mishchenko. Involution of manifolds with a set of fixed points diffeomorphic to real projective space. Matematičeskie zametki, Tome 9 (1971) no. 3, pp. 249-252. http://geodesic.mathdoc.fr/item/MZM_1971_9_3_a2/