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On Manifolds with the Homotopy Type of Complex Projective Space

Bruce Conrad
Transactions of the American Mathematical Society
Vol. 176 (Feb., 1973), pp. 165-180
DOI: 10.2307/1996202
Stable URL: http://www.jstor.org/stable/1996202
Page Count: 16
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On Manifolds with the Homotopy Type of Complex Projective Space
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Abstract

It is known that in every even dimension greater than four there are infinitely many nonhomeomorphic smooth manifolds with the homotopy type of complex projective space. In this paper we provide an explicit construction of homotopy complex projective spaces. Our initial data will be a manifold X with the homotopy type of CP3 and an embedding γ3: S5 → S7. A homotopy 7-sphere Σ7 is constructed and an embedding γ4: Σ7 → S9 may be chosen. The procedure continues inductively until either an obstruction or the desired dimension is reached; in the latter case the final obstruction is the class of Σ2n - 1 in Θ2n - 1. Should this obstruction vanish, the final choice is of a diffeomorphism γn: Σ2n - 1 → S2n - 1. There results a manifold, denoted (X, γ3,⋯, γn - 1, γn), with the homotopy type of CPn. We describe the obstructions encountered, but are able to evaluate only the primary ones. It is shown that every homotopy complex projective space may be so constructed, and in terms of this construction, necessary and sufficient conditions for two homotopy complex projective spaces to be diffeomorphic are stated.

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