## Manifolds with finite first homology as codimension $2$ fibrators

HTML articles powered by AMS MathViewer

- by Robert J. Daverman PDF
- Proc. Amer. Math. Soc.
**113**(1991), 471-477 Request permission

## Abstract:

Given a map $f:M \to B$ defined on an orientable $(n + 2)$-manifold with all point inverses having the homotopy type of a specified closed $n$-manifold $N$, we seek to catalog the manifolds $N$ for which $f$ is always an approximate fibration. Assuming ${H_1}(N)$ finite, we deduce that the cohomology sheaf of $f$ is locally constant provided $N$ admits no self-map of degree $d > 1$ when ${H_1}(N)$ has a cyclic subgroup of order $d$. For manifolds $N$ possessing additional features, we achieve the approximate fibration conclusion.## References

- D. S. Coram and P. F. Duvall Jr.,
*Approximate fibrations*, Rocky Mountain J. Math.**7**(1977), no. 2, 275–288. MR**442921**, DOI 10.1216/RMJ-1977-7-2-275 - Donald Coram and Paul Duvall,
*Approximate fibrations and a movability condition for maps*, Pacific J. Math.**72**(1977), no. 1, 41–56. MR**467745** - R. J. Daverman,
*Submanifold decompositions that induce approximate fibrations*, Topology Appl.**33**(1989), no. 2, 173–184. MR**1020279**, DOI 10.1016/S0166-8641(89)80006-9 - R. J. Daverman and J. J. Walsh,
*Decompositions into codimension-two manifolds*, Trans. Amer. Math. Soc.**288**(1985), no. 1, 273–291. MR**773061**, DOI 10.1090/S0002-9947-1985-0773061-4 - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - Jean-Claude Hausmann,
*Geometric Hopfian and non-Hopfian situations*, Geometry and topology (Athens, Ga., 1985) Lecture Notes in Pure and Appl. Math., vol. 105, Dekker, New York, 1987, pp. 157–166. MR**873292** - John Hempel,
*$3$-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619**
—, - Peter Scott,
*The geometries of $3$-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, DOI 10.1112/blms/15.5.401 - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

*Residual finiteness for*$3$

*-manifolds*, Combinatorial Group Theory and Topology (S. M. Gersten and J. R. Stallings, eds.), Ann. of Math. Stud., No. 111, Princeton Univ. Press, 1987, pp. 379-396.

## Additional Information

- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**113**(1991), 471-477 - MSC: Primary 55R65; Secondary 54B15, 57M25, 57N12, 57N15, 57N65
- DOI: https://doi.org/10.1090/S0002-9939-1991-1086581-0
- MathSciNet review: 1086581