Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some mapping theorems


Author: R. C. Lacher
Journal: Trans. Amer. Math. Soc. 195 (1974), 291-303
MSC: Primary 57A15
MathSciNet review: 0350743
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Various mapping theorems are proved, culminating in the following result for mappings f from a closed $ (2k + 1)$-manifold M to another, N: If ``almost all'' point-inverses of f are strongly acyclic in dimensions less than k and if ``almost all'' point-inverses of f have Euler characteristic equal to one, then all but finitely many point-inverses are totally acyclic. (Here ``almost all'' means ``except on a zero-dimensional set in N".) More can be said when $ k = 1$: If f is a monotone map between closed 3-manifolds and if the Euler characteristic of almost-all point-inverses is one, then all but finitely many point-inverses of f are cellular in M; consequently M is the connected sum of N and some other closed 3-manifold and f is homotopic to a spine map. Other results include an acyclicity criterion using the idea of ``nonalternating'' mapping and the following result for PL maps $ \phi $ between finite polyhedra X and Y: If the Euler characteristic of each point-inverse of $ \phi $ is the integer c then $ \chi (X) = c\chi (Y)$.


References [Enhancements On Off] (What's this?)

  • [1] S. Armentrout, UV-properties of compact sets, Trans. Amer. Math. Soc. 143 (1969), 487-498. MR 42 #8451. MR 0273573 (42:8451)
  • [2] -, Cellular decompositions of 3-manifolds that yield 3-manifolds, Mem. Amer. Math. Soc. No. 107 (1971). MR 0413104 (54:1225)
  • [3] E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math. (2) 51 (1950), 534-543. MR 11, 677. MR 0035015 (11:677b)
  • [4] R. H. Bing, The monotone mapping problem, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 99-115. MR 43 #1136. MR 0275379 (43:1136)
  • [5] G. E. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 36 #4552. MR 0221500 (36:4552)
  • [6] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [7] R. C. Lacher, Cellularity criteria for maps, Michigan Math. J. 17 (1970), 385-396. MR 43 #5539. MR 0279818 (43:5539)
  • [8] -, Finiteness theorems in the study of mappings between manifolds, Proc. Conf. Top. (Univ. of Oklahoma, 1972), Dept. of Math., University of Oklahoma, Norman, 1972, pp. 79-96. MR 0370593 (51:6820)
  • [9] R. C. Lacher and D. R. McMillan, Jr., Partially acyclic mappings between manifolds, Amer. J. Math. 94 (1972), 246-266. MR 46 #898. MR 0301743 (46:898)
  • [10] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-294. MR 45 #4431. MR 0295365 (45:4431)
  • [11] E. G. Skljarenko, Almost acyclic mappings, Mat. Sb. 75 (117) (1968), 296-302 = Math. USSR Sb. 4 (1968), 267-272. MR 37 #4806. MR 0229232 (37:4806)
  • [12] -, Homology theory and the exactness axiom, Uspehi Mat. Nauk 24 (1969), no. 5 (149), 87-140 = Russian Math. Surveys 24 (1969), no. 5, 91-142. MR 41 #7676. MR 0263071 (41:7676)
  • [13] R. Soloway, Somewhere acyclic mappings of manifolds are compact, Ph.D. Thesis, University of Wisconsin, 1971.
  • [14] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [15] D. Sullivan, Combinatorial invariants of analytic spaces, Proc. of Liverpool Singularities-Sympos., I (1969/70), Lecture Notes in Math., vol. 192, Springer, Berlin, 1971, pp. 165-168. MR 43 #4063. MR 0278333 (43:4063)
  • [16] 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)
  • [17] A. H. Wright, Mappings from 3-manifolds onto 3-manifolds, Trans. Amer. Math. Soc. 167 (1972), 479-495. MR 0339186 (49:3949)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57A15

Retrieve articles in all journals with MSC: 57A15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0350743-2
PII: S 0002-9947(1974)0350743-2
Keywords: Mapping, acyclic, finiteness, cellularity
Article copyright: © Copyright 1974 American Mathematical Society