Differentiable monotone maps on manifolds. II
Author:
P. T. Church
Journal:
Trans. Amer. Math. Soc. 158 (1971), 493-501
MSC:
Primary 57.20
DOI:
https://doi.org/10.1090/S0002-9947-1971-0278320-X
MathSciNet review:
0278320
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let and
be closed manifolds, and let
be any (nonzero) module. (1) If
is
-acyclic, then there is a closed
-manifold
such that
is diffeomorphic to
, and
is cellular for all but at most
points
, where
is the number of nontrivial
-cohomology
-spheres in the prime decomposition of
. (2) If
or
is
-acyclic, then
is cellular. In case
is
or
(
prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If
or
is real analytic monotone onto, then
is a homeomorphism.
- [1]
S. Armentrout, Cellular decompositions of
-manifolds that yield
-manifolds, Bull. Amer. Math. Soc. 75 (1969), 453-456. MR 39 #935. MR 0239578 (39:935)
- [2]
-, Upper semi-continuous decompositions of
with at most countably many nondegenerate elements, Ann. of Math. (2) 78 (1963), 605-618. MR 27 #6255. MR 0156331 (27:6255)
- [3] A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513. MR 26 #6990. MR 0149503 (26:6990)
- [4] M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 22 #8470b. MR 0117695 (22:8470b)
- [5] P. T. Church, Differentiable maps with non-negative Jacobian, J. Math. Mech. 16 (1966), 703-708. MR 34 #5096. MR 0205263 (34:5096)
- [6] -, Differentiable monotone maps on manifolds, Trans. Amer. Math. Soc. 128 (1967), 185-205. MR 39 #3505. MR 0242172 (39:3505)
- [7] -, Differentiable open maps on manifolds, Trans. Amer. Math. Soc. 109 (1963), 87-100. MR 27 #4245. MR 0154296 (27:4245)
- [8]
-, Factorization of differentiable maps with branch set dimension at most
, Trans. Amer. Math. Soc. 115 (1965), 370-387. MR 33 #3306. MR 0195101 (33:3306)
- [9]
-, On points of Jacobian rank
. II, Proc. Amer. Math. Soc. 16 (1965), 1035-1038. MR 31 #6241. MR 0182017 (31:6241)
- [10] P. T. Church and W. D. Nathan, Real analytic maps on manifolds, J. Math. Mech. 19 (1969), 19-36. MR 39 #4860. MR 0243539 (39:4860)
- [11] P. T. Church and J. G. Timourian, Fiber bundles with singularities, J. Math. Mech. 18 (1968/69), 71-90. MR 38 #3876. MR 0235572 (38:3876)
- [12] H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jber. Deutsch. Math.-Verein. 30 (1929), 248-260.
- [13] K. W. Kwun and F. Raymond, Almost acyclic maps of manifolds, Amer. J. Math. 86 (1964), 638-649. MR 32 #1712. MR 0184239 (32:1712)
- [14] S. MacLane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122. MR 0156879 (28:122)
- [15] D. R. McMillan, Jr., Acyclicity in three-manifolds, Bull. Amer. Math. Soc. 76(1970), 942-964. MR 0270380 (42:5269)
- [16] -, Compact, acyclic subsets of three-manifolds, Michigan Math. J. 16 (1969), 129136. MR 39 #4822. MR 0243501 (39:4822)
- [17] -, Decompositions of euclidean space yielding a manifold, Notices Amer. Math. Soc. 17 (1970), 679. Abstract #70T-G81.
- [18]
-, Strong homotopy equivalence of
-manifolds, Bull. Amer. Math. Soc. 73 (1967), 718-722. MR 37 #4817. MR 0229243 (37:4817)
- [19]
J. Milnor, A unique decomposition theorem for
-manifolds, Amer. J. Math. 84 (1962), 1-7. MR 25 #5518. MR 0142125 (25:5518)
- [20] J. R. Munkres, Elementary differential topology, Ann. of Math. Studies, no. 54, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #623. MR 0163320 (29:623)
- [21] -, Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. of Math. (2) 72 (1960), 521-554. MR 22 #12534. MR 0121804 (22:12534)
- [22] R. S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125-141. MR 22 #7140. MR 0116352 (22:7140)
- [23] T. Radó and P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der math. Wissenschaften, Band 65, SpringerVerlag, Berlin, 1955. MR 18, 115. MR 0079620 (18:115c)
- [24] J. H. Roberts and N. E. Steenrod, Monotone transformations of two-dimensional manifolds, Ann. of Math. (2) 39 (1938), 851-862. MR 1503441
- [25] A. Sard, Hausdorff measure of critical images on Banach manifolds, Amer. J. Math. 87 (1965), 158-174. MR 30 #3958. MR 0173748 (30:3958)
- [26] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Notices Amer. Math. Soc. 17 (1970), 532. Abstract #674-40.
- [27] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
- [28] S. Sternberg, Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 33 #1797. MR 0193578 (33:1797)
- [29] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. MR 15, 890. MR 0061823 (15:890a)
- [30] J. R. Walker, Monotone mappings and monotone decompositions, Ph.D. Dissertation, Syracuse University, Syracuse, New York, 1970.
- [31] H. Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), 645-680. MR 1503303
- [32] G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I;, 1942. MR 4, 86. MR 0007095 (4:86b)
- [33]
A. Wright, Mappings from
-manifolds onto
-manifolds (to appear).
- [34]
-, Monotone mappings of compact
-manifolds, Notices Amer. Math. Soc. 16 (1969), 853-854. Abstract #69T-G121.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1971-0278320-X
Keywords:
Monotone,
acyclic,
cellular,
open,
differentiable,
real analytic maps
Article copyright:
© Copyright 1971
American Mathematical Society