Geodesic
On the topological structure of compact 5-manifolds
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 513-524

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We classify the genus one compact (PL) 5-manifolds and prove some results about closed 5-manifolds with free fundamental group. In particular, let $M$ be a closed connected orientable smooth $5$-manifold with free fundamental group. Then we prove that the number of distinct smooth $5$-manifolds homotopy equivalent to $M$ equals the $2$-nd Betti number (mod $2$) of $M$.
Classification : 57N15, 57N65, 57Q25, 57R67
Keywords: colored graph; crystallization; genus; manifold; surgery; s-cobordism; normal invariants; homotopy type 
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Cavicchioli, Alberto; Spaggiari, Fulvia. On the topological structure of compact 5-manifolds. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 513-524. http://geodesic.mathdoc.fr/item/CMUC_1993__34_3_a12/