A controlled plus construction for crumpled laminations
Authors:
R. J. Daverman and F. C. Tinsley
Journal:
Trans. Amer. Math. Soc. 342 (1994), 807826
MSC:
Primary 57N70; Secondary 54B15, 57M20, 57N35
MathSciNet review:
1182981
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given a closed nmanifold M and a finitely generated perfect subgroup P of , we previously developed a controlled version of Quillen's plus construction, namely a cobordism (W, M, N) with the inclusion a homotopy equivalence and kernel of equalling the smallest normal subgroup of containing P together with a closed map such that is a closed nmanifold for every and, in particular, and . We accomplished this by constructing an acyclic map of manifolds having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when contains no finitely generated perfect subgroups. Byproducts of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.
 [Ad]
J.
F. Adams, A new proof of a theorem of W. H. Cockcroft, J.
London Math. Soc. 30 (1955), 482–488. MR 0076335
(17,883d)
 [Ar]
Steve
Armentrout, Decompositions and absolute neighborhood retracts,
Geometric topology (Proc. Conf., Park City, Utah, 1974) Springer, Berlin,
1975, pp. 1–5. Lecture Notes in Math., Vol. 438. MR 0394600
(52 #15401)
 [B]
R.
H. Bing, Upper semicontinuous decompositions of
𝐸³, Ann. of Math. (2) 65 (1957),
363–374. MR 0092960
(19,1187f)
 [C1]
J.
W. Cannon, Shrinking celllike decompositions of manifolds.
Codimension three, Ann. of Math. (2) 110 (1979),
no. 1, 83–112. MR 541330
(80j:57013), http://dx.doi.org/10.2307/1971245
 [C2]
J.
W. Cannon, The recognition problem: what is a
topological manifold?, Bull. Amer. Math.
Soc. 84 (1978), no. 5, 832–866. MR 0494113
(58 #13043), http://dx.doi.org/10.1090/S000299041978145273
 [CBL]
J.
W. Cannon, J.
L. Bryant, and R.
C. Lacher, The structure of generalized manifolds having
nonmanifold set of trivial dimension, Geometric topology (Proc.
Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New
YorkLondon, 1979, pp. 261–300. MR 537735
(80h:57026)
 [Da1]
Robert
J. Daverman, Every crumpled 𝑛cube is a closed
𝑛cellcomplement, Michigan Math. J. 24
(1977), no. 2, 225–241. MR 0488066
(58 #7637)
 [Da2]
Robert
J. Daverman, Detecting the disjoint disks property, Pacific J.
Math. 93 (1981), no. 2, 277–298. MR 623564
(82k:57007)
 [Da3]
R.
J. Daverman, Decompositions of manifolds into codimension one
submanifolds, Compositio Math. 55 (1985), no. 2,
185–207. MR
795714 (87b:57016)
 [Da4]
Robert
J. Daverman, Decompositions of manifolds, Pure and Applied
Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. MR 872468
(88a:57001)
 [DT1]
R.
J. Daverman and F.
C. Tinsley, Laminated decompositions involving a given
submanifold, Topology Appl. 20 (1985), no. 2,
107–119. MR
800841 (87d:57013), http://dx.doi.org/10.1016/01668641(85)900719
 [DT2]
R.
J. Daverman and F.
C. Tinsley, Laminations, finitely generated perfect groups, and
acyclic maps, Michigan Math. J. 33 (1986),
no. 3, 343–351. MR 856526
(87k:57016), http://dx.doi.org/10.1307/mmj/1029003414
 [DT3]
R.
J. Daverman and F.
C. Tinsley, The homotopy type of certain laminated
manifolds, Proc. Amer. Math. Soc.
96 (1986), no. 4,
703–708. MR
826506 (87e:57024), http://dx.doi.org/10.1090/S00029939198608265061
 [DW1]
R.
J. Daverman and J.
J. Walsh, Decompositions into codimension two spheres and
approximate fibrations, Topology Appl. 19 (1985),
no. 2, 103–121. MR 789592
(87h:57020), http://dx.doi.org/10.1016/01668641(85)900641
 [DW2]
R.
J. Daverman and J.
J. Walsh, Decompositions into codimensiontwo
manifolds, Trans. Amer. Math. Soc.
288 (1985), no. 1,
273–291. MR
773061 (87h:57019), http://dx.doi.org/10.1090/S00029947198507730614
 [DW3]
R.
J. Daverman and J.
J. Walsh, Decompositions into submanifolds that yield generalized
manifolds, Topology Appl. 26 (1987), no. 2,
143–162. MR
896870 (89c:57017), http://dx.doi.org/10.1016/01668641(87)900654
 [Do]
A.
Dold, Lectures on algebraic topology, SpringerVerlag, New
YorkBerlin, 1972 (German). Die Grundlehren der mathematischen
Wissenschaften, Band 200. MR 0415602
(54 #3685)
 [Dr]
A.
N. Dranishnikov, On a problem of P. S. Aleksandrov, Mat. Sb.
(N.S.) 135(177) (1988), no. 4, 551–557, 560
(Russian); English transl., Math. USSRSb. 63 (1989),
no. 2, 539–545. MR 942139
(90e:55004)
 [E1]
Robert
D. Edwards, Demension theory. I, Geometric topology (Proc.
Conf., Park City, Utah, 1974) Springer, Berlin, 1975,
pp. 195–211. Lecture Notes in Math., Vol. 438. MR 0394678
(52 #15477)
 [E2]
Robert
D. Edwards, The topology of manifolds and celllike maps,
Proceedings of the International Congress of Mathematicians (Helsinki,
1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 111–127. MR 562601
(81g:57010)
 [F]
Michael
Hartley Freedman, The topology of fourdimensional manifolds,
J. Differential Geom. 17 (1982), no. 3,
357–453. MR
679066 (84b:57006)
 [H]
James
Howie, Aspherical and acyclic 2complexes, J. London Math.
Soc. (2) 20 (1979), no. 3, 549–558. MR 561147
(81e:57004), http://dx.doi.org/10.1112/jlms/s220.3.549
 [Li]
Vo
Thanh Liem, Manifolds accepting codimensionone sphereshape
decompositions, Topology Appl. 21 (1985), no. 1,
77–86. MR
808726 (87a:57022), http://dx.doi.org/10.1016/01668641(85)900604
 [LS]
Roger
C. Lyndon and Paul
E. Schupp, Combinatorial group theory, Classics in
Mathematics, SpringerVerlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024
(2001i:20064)
 [Q1]
Daniel
Quillen, Cohomology of groups, Actes du Congrès
International des Mathématiciens (Nice, 1970) GauthierVillars,
Paris, 1971, pp. 47–51. MR 0488054
(58 #7627a)
 [Qn1]
Frank
Quinn, Ends of maps. I, Ann. of Math. (2) 110
(1979), no. 2, 275–331. MR 549490
(82k:57009), http://dx.doi.org/10.2307/1971262
 [Qn2]
Frank
Quinn, Resolutions of homology manifolds, and the topological
characterization of manifolds, Invent. Math. 72
(1983), no. 2, 267–284. MR 700771
(85b:57023), http://dx.doi.org/10.1007/BF01389323
 [Qn3]
Frank
Quinn, An obstruction to the resolution of homology manifolds,
Michigan Math. J. 34 (1987), no. 2, 285–291. MR 894878
(88j:57016), http://dx.doi.org/10.1307/mmj/1029003559
 [Si]
L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than 5, Ph.D. thesis, Princeton Univ., 1965.
 [Sm]
Stephen
Smale, A Vietoris mapping theorem for
homotopy, Proc. Amer. Math. Soc. 8 (1957), 604–610. MR 0087106
(19,302f), http://dx.doi.org/10.1090/S00029939195700871069
 [T]
F. C. Tinsley, Acyclic maps which are homotopic to homeomorphisms, Abstract #8385731, Abstracts Amer. Math. Soc. 8 (1987), p. 426.
 [W]
C.
T. C. Wall (ed.), Homological group theory, London
Mathematical Society Lecture Note Series, vol. 36, Cambridge
University Press, CambridgeNew York, 1979. MR 564417
(80m:20001)
 [Ad]
 J. F. Adams, A new proof of a theorem of W. H. Cockcroft, J. London Math. Soc. 49 (1955), 482488. MR 0076335 (17:883d)
 [Ar]
 S. Armentrout, Decompositions and absolute neighborhood retracts, Geometric Topology (L. C. Glaser and T. B. Rushing, eds.), Lecture Notes in Math., vol. 438, SpringerVerlag, Berlin and New York, 1975, pp. 15. MR 0394600 (52:15401)
 [B]
 R. H. Bing, Upper semicontinuous decompositions of , Ann. of Math. (2) 65 (1957), 363374. MR 0092960 (19:1187f)
 [C1]
 J. W. Cannon, Shrinking celllike decompositions of manifolds. Codimension three, Ann. of Math. (2) 110 (1979), 83112. MR 541330 (80j:57013)
 [C2]
 , The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), 832866. MR 0494113 (58:13043)
 [CBL]
 J. W. Cannon, J. L. Bryant, and R. C. Lacher, The structure of generalized manifolds having nonmanifold set of trivial dimension, Geometric Topology (J. C. Cantrell, ed.), Academic Press, New York, 1979, pp. 261300. MR 537735 (80h:57026)
 [Da1]
 R. J. Daverman, Every crumpled ncube is a closed 4cellcomplement, Michigan Math. J. 24 (1977), 225241. MR 0488066 (58:7637)
 [Da2]
 , Detecting the disjoint disks property, Pacific J. Math. 93 (1981), 277298. MR 623564 (82k:57007)
 [Da3]
 , Decompositions into codimension one submanifolds, Compositio Math. 55 (1985), 185207. MR 795714 (87b:57016)
 [Da4]
 , Decompositions of manifolds, Academic Press, Orlando, 1986. MR 872468 (88a:57001)
 [DT1]
 R. J. Daverman and F. C. Tinsley, Laminated decompositions involving a given submanifold, Topology Appl. 20 (1985), 107119. MR 800841 (87d:57013)
 [DT2]
 , Laminations, finitely generated perfect groups, and acyclic mappings, Michigan Math. J. 33 (1986), 343351. MR 856526 (87k:57016)
 [DT3]
 , The homotopy type of certain laminated manifolds, Proc. Amer. Math. Soc. 96 (1986), 703708. MR 826506 (87e:57024)
 [DW1]
 R. J. Daverman and J. J. Walsh, Decompositions into codimension two spheres and approximate fibrations, Topology Appl. 19 (1985), 103121. MR 789592 (87h:57020)
 [DW2]
 , Decompositions into codimension two manifolds, Trans. Amer. Math. Soc. 288 (1985), 273291. MR 773061 (87h:57019)
 [DW3]
 , Decompositions into submanifolds that yield generalized manifolds, Topology Appl. 26 (1987), 143162. MR 896870 (89c:57017)
 [Do]
 Albrecht Dold, Lectures on algebraic topology, SpringerVerlag, Berlin, Heidelberg, and New York, 1972. MR 0415602 (54:3685)
 [Dr]
 A. N. Dranishnikov, On a problem of P. S. Aleksandrov, Math. USSRSb. 63 (21) (1989), 539545; English transl. of Mat. Sb. 135 (177) (1988), 551557. MR 942139 (90e:55004)
 [E1]
 R. D. Edwards, Demension theory. I, Geometric Topology (L. C. Glaser and T. B. Rushing, eds.), Lecture Notes in Math., vol. 438, SpringerVerlag, Berlin and New York, 1975, pp. 195211. MR 0394678 (52:15477)
 [E2]
 , The topology of manifolds and celllike maps, Proc. Internat. Congr. Math. Helsinki, 1978 (O. Lehti, ed.), Acad. Sci. Fenn., Helsinki, 1980, pp. 111127. MR 562601 (81g:57010)
 [F]
 M. H. Freedman, The topology of fourdimensional manifolds, J. Differential Geom. 17 (1982), 352453. MR 679066 (84b:57006)
 [H]
 J. Howie, Aspherical and acyclic 2complexes, J. London Math. Soc. (2) 20 (1979), 549558. MR 561147 (81e:57004)
 [Li]
 V. T. Liem, Manifolds accepting codimension one spherelike decompositions, Topology Appl. 21 (1985), 7786. MR 808726 (87a:57022)
 [LS]
 R. C. Lyndon and P. E. Schupp, Combinatorial group theory, SpringerVerlag, Berlin and New York, 1976. MR 1812024 (2001i:20064)
 [Q1]
 D. Quillen, Cohomology of groups, Actes Congres Int. Math., Tome 2, 1970, pp. 4751. MR 0488054 (58:7627a)
 [Qn1]
 F. Quinn, Ends of maps. I, Ann. of Math. (2) 110 (1979), 275331. MR 549490 (82k:57009)
 [Qn2]
 , Resolutions of manifolds, and the topological characterization of manifolds, Invent. Math. 72 (1983), 267284. MR 700771 (85b:57023)
 [Qn3]
 , An obstruction to the resolution of homology manifolds, Michigan Math. J. 34 (1987), 285291. MR 894878 (88j:57016)
 [Si]
 L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than 5, Ph.D. thesis, Princeton Univ., 1965.
 [Sm]
 S. Smale, A Vietoris theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604610. MR 0087106 (19:302f)
 [T]
 F. C. Tinsley, Acyclic maps which are homotopic to homeomorphisms, Abstract #8385731, Abstracts Amer. Math. Soc. 8 (1987), p. 426.
 [W]
 C. T. C. Wall (Editor), Homological group theory, Cambridge Univ. Press, London, 1979. MR 564417 (80m:20001)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
57N70,
54B15,
57M20,
57N35
Retrieve articles in all journals
with MSC:
57N70,
54B15,
57M20,
57N35
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199411829816
PII:
S 00029947(1994)11829816
Keywords:
Crumpled lamination,
degree one map,
almost acyclic,
perfect normal subgroup
Article copyright:
© Copyright 1994
American Mathematical Society
