The Poincaré conjecture is true in the product of any graph with a disk
Abstract: We prove that the only compact -manifold-with-boundary which has trivial rational homology, and which embeds in the product of a graph with a disk, is the -ball. This implies that no punctured lens space embeds in the product of a graph with a disk. It also implies our title.
The proof relies on a general position argument which enables us to perform surgery.
-  D. B. A. Epstein, Embedding punctured manifolds, Proc. Amer. Math. Soc. 16 (1965), 175–176. MR 0208606, 10.1090/S0002-9939-1965-0208606-9
-  D. Gillman and D. Rolfsen, The Zeeman conjecture for standard spines is equivalent to the Poincaré conjecture, Topology 22 (1983), no. 3, 315–323. MR 710105, 10.1016/0040-9383(83)90017-4
-  Edwin E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 47. MR 0488059
-  Herbert Seifert and William Threlfall, Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, vol. 89, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman; With a preface by Joan S. Birman; With “Topology of 3-dimensional fibered spaces” by Seifert; Translated from the German by Wolfgang Heil. MR 575168
-  E. C. Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471–495. MR 0195085, 10.1090/S0002-9947-1965-0195085-8
- D. B. A. Epstein, Embedding punctured manifolds, Proc. Amer. Math. Soc. 16 (1965), 175-176. MR 0208606 (34:8415)
- D. Gillman and D. Rolfsen, The Zeeman Conjecture for standard spines is equivalent to the Poincaré Conjecture, Topology 22 (1983), 315-323. MR 710105 (85b:57013)
- E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York, 1977. MR 0488059 (58:7631)
- H. Seifert and W. Threlfall, A textbook of topology, Academic Press, New York, 1980. MR 575168 (82b:55001)
- E. C. Zeeman, On twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. MR 0195085 (33:3290)