Some mapping theorems

Author:
R. C. Lacher

Journal:
Trans. Amer. Math. Soc. **195** (1974), 291-303

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'' 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 : 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 between finite polyhedra *X* and *Y*: If the Euler characteristic of each point-inverse of is the integer *c* then .

**[1]**Steve Armentrout,*𝑈𝑉 properties of compact sets*, Trans. Amer. Math. Soc.**143**(1969), 487–498. MR**0273573**, 10.1090/S0002-9947-1969-0273573-7**[2]**Steve Armentrout,*Cellular decompositions of 3-manifolds that yield 3-manifolds*, American Mathematical Society, Providence, R. I., 1971. Memoirs of the American Mathematical Society, No. 107. MR**0413104****[3]**Edward G. Begle,*The Vietoris mapping theorem for bicompact spaces*, Ann. of Math. (2)**51**(1950), 534–543. MR**0035015****[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**0275379****[5]**Glen E. Bredon,*Sheaf theory*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR**0221500****[6]**Henri Cartan and Samuel Eilenberg,*Homological algebra*, Princeton University Press, Princeton, N. J., 1956. MR**0077480****[7]**R. C. Lacher,*Cellularity criteria for maps*, Michigan Math. J.**17**(1970), 385–396. MR**0279818****[8]**R. C. Lacher,*Finiteness theorems in the study of mappings between manifolds*, Proceedings of the University of Oklahoma Topology Conference Dedicated to Robert Lee Moore (Norman, Okla., 1972) Univ. Oklahoma, Norman, Okla., 1972, pp. 79–96. MR**0370593****[9]**R. C. Lacher and D. R. McMillan Jr.,*Partially acyclic mappings between manifolds*, Amer. J. Math.**94**(1972), 246–266. MR**0301743****[10]**L. C. Siebenmann,*Approximating cellular maps by homeomorphisms*, Topology**11**(1972), 271–294. MR**0295365****[11]**E. G. Skljarenko,*Almost acyclic mappings*, Mat. Sb. (N.S.)**75 (117)**(1968), 296–302 (Russian). MR**0229232****[12]**E. G. Skljarenko,*Homology theory and the exactness axiom*, Uspehi Mat. Nauk**24**(1969), no. 5 (149), 87–140 (Russian). MR**0263071****[13]**R. Soloway,*Somewhere acyclic mappings of manifolds are compact*, Ph.D. Thesis, University of Wisconsin, 1971.**[14]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[15]**D. Sullivan,*Combinatorial invariants of analytic spaces*, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Springer, Berlin, 1971, pp. 165–168. MR**0278333****[16]**Gordon Thomas Whyburn,*Analytic Topology*, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR**0007095****[17]**Alden Wright,*Mappings from 3-manifolds onto 3-manifolds*, Trans. Amer. Math. Soc.**167**(1972), 479–495. MR**0339186**, 10.1090/S0002-9947-1972-0339186-3

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DOI:
https://doi.org/10.1090/S0002-9947-1974-0350743-2

Keywords:
Mapping,
acyclic,
finiteness,
cellularity

Article copyright:
© Copyright 1974
American Mathematical Society