# Article

Full entry | PDF   (0.2 MB)
Keywords:
colored graph; crystallization; genus; manifold; surgery; s-cobordism; normal invariants; homotopy type
Summary:
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 $M$.
References:
[1] Barden D.: Simply connected five-manifolds. Ann. of Math. 82 (1965), 365-385. MR 0184241 | Zbl 0136.20602
[2] Bracho J., Montejano L.: The combinatorics of colored triangulations of manifolds. Geom. Dedicata 22 (1987), 303-328. MR 0887580 | Zbl 0631.57017
[3] Cappell S.: Mayer-Vietoris sequences in hermitian $K$-theory. preprint. MR 0358814 | Zbl 0298.57021
[4] Cavicchioli A.: A combinatorial characterization of $\Bbb S^3\times \Bbb S^1$ among closed 4-manifolds. Proc. Amer. Math. Soc. 105 (1989), 1008-1014. MR 0931726
[5] Ferri M., Gagliardi C., Grasselli L.: A graph-theoretical representation of PL-manifolds. A survey on crystallizations. Aequationes Math. 31 (1986), 121-141. MR 0867510 | Zbl 0623.57012
[6] Mandelbaum R.: Four-dimensional topology: an introduction. Bull. Amer. Math. Soc. 2 (1980), 1-159. MR 0551752 | Zbl 0476.57005
[7] Milnor J.W.: A procedure for killing the homotopy groups of differentiable manifolds. in Proc. Symp. in Pure Math. (Differential Geometry), Amer. Math. Soc. 3 (1961), 39-55. MR 0130696
[8] Milnor J.: Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 0196736 | Zbl 0147.23104
[9] Rourke C.P., Sanderson B.J.: Introduction to piecewise-linear topology. Springer-Verlag Ed., Berlin-Heidelberg-New York, 1972. MR 0350744 | Zbl 0477.57003
[10] Shaneson J.L.: Wall's surgery obstruction groups for $G\times \Bbb Z$. Ann. of Math. 90 (1969), 296-334. MR 0246310
[11] Shaneson J.L.: Non-simply connected surgery and some results in low dimension topology. Comm. Math. Helv. 45 (1970), 333-352. MR 0275444
[12] Shaneson J.L.: On non-simply connected manifolds. in Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, Rhode Island 22 (1970), 221-229. MR 0358816 | Zbl 0226.57007
[13] Smale S.: On the structure of 5-manifolds. Ann. of Math. 75 (1962), 38-46. MR 0141133 | Zbl 0101.16103
[14] Wall C.T.C.: Surgery on Compact Manifolds. Academic Press, London-New York, 1970. MR 0431216 | Zbl 0935.57003
[15] White A.T.: Graphs, Groups and Surfaces. North Holland Ed., Amsterdam, 1973. Zbl 0551.05037

Partner of