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

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    $\begingroup$ 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. $\endgroup$
    – Bruno Martelli
    Nov 3, 2011 at 17:33

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Such a thing is either Haken, or geometric (by Geometrization). Waldhausen showed that universal covers of Haken manifolds are cells, and you get a sphere or cell universal covering in the geometric case for free.

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