# is very compact P^2-irreducible 3-manifold homotopy equivalent to a sphere or cell-quotient?

is very compact $P^2$-irreducible 3-manifold homotopy equivalent to a sphere or cell-quotient?

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What do you mean by "cell-quotient"? If you mean a quotient of the three-cell via maps that identify points of its boundary, then every 3-manifold is homeomorphic to such an object. –  Bruno Martelli Nov 3 '11 at 17:33