Some mapping theorems
Author:
R. C. Lacher
Journal:
Trans. Amer. Math. Soc. 195 (1974), 291303
MSC:
Primary 57A15
MathSciNet review:
0350743
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Abstract: Various mapping theorems are proved, culminating in the following result for mappings f from a closed manifold M to another, N: If ``almost all'' pointinverses of f are strongly acyclic in dimensions less than k and if ``almost all'' pointinverses of f have Euler characteristic equal to one, then all but finitely many pointinverses are totally acyclic. (Here ``almost all'' means ``except on a zerodimensional set in N".) More can be said when : If f is a monotone map between closed 3manifolds and if the Euler characteristic of almostall pointinverses is one, then all but finitely many pointinverses of f are cellular in M; consequently M is the connected sum of N and some other closed 3manifold 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 between finite polyhedra X and Y: If the Euler characteristic of each pointinverse of is the integer c then .
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 [1]
 S. Armentrout, UVproperties of compact sets, Trans. Amer. Math. Soc. 143 (1969), 487498. MR 42 #8451. MR 0273573 (42:8451)
 [2]
 , Cellular decompositions of 3manifolds that yield 3manifolds, 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), 534543. 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. 99115. MR 43 #1136. MR 0275379 (43:1136)
 [5]
 G. E. Bredon, Sheaf theory, McGrawHill, 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), 385396. 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. 7996. MR 0370593 (51:6820)
 [9]
 R. C. Lacher and D. R. McMillan, Jr., Partially acyclic mappings between manifolds, Amer. J. Math. 94 (1972), 246266. MR 46 #898. MR 0301743 (46:898)
 [10]
 L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271294. MR 45 #4431. MR 0295365 (45:4431)
 [11]
 E. G. Skljarenko, Almost acyclic mappings, Mat. Sb. 75 (117) (1968), 296302 = Math. USSR Sb. 4 (1968), 267272. MR 37 #4806. MR 0229232 (37:4806)
 [12]
 , Homology theory and the exactness axiom, Uspehi Mat. Nauk 24 (1969), no. 5 (149), 87140 = Russian Math. Surveys 24 (1969), no. 5, 91142. 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, McGrawHill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
 [15]
 D. Sullivan, Combinatorial invariants of analytic spaces, Proc. of Liverpool SingularitiesSympos., I (1969/70), Lecture Notes in Math., vol. 192, Springer, Berlin, 1971, pp. 165168. 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 3manifolds onto 3manifolds, Trans. Amer. Math. Soc. 167 (1972), 479495. MR 0339186 (49:3949)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403507432
PII:
S 00029947(1974)03507432
Keywords:
Mapping,
acyclic,
finiteness,
cellularity
Article copyright:
© Copyright 1974
American Mathematical Society
