The closure of the space of homeomorphisms on a manifold
Author:
William E. Haver
Journal:
Trans. Amer. Math. Soc. 195 (1974), 401419
MSC:
Primary 57E05; Secondary 57A20
MathSciNet review:
0362375
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The space, , of all mappings of the compact manifold M onto itself which can be approximated arbitrarily closely by homeomorphisms is studied. It is shown that is homogeneous and weakly locally contractible. If M is a compact 2manifold without boundary, then is shown to be locally contractible.
 [1]
J. W. Alexander, On the deformation of an ncell, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 406407.
 [2]
R. D. Anderson, Strongly negligible sets in Fréchet manifolds, Bull. Amer. Math. Soc. 75 (1969), 6467. MR 38 #6634. MR 0238358 (38:6634)
 [3]
, Spaces of homeomorphisms of finite graphs (to appear).
 [4]
S. Armentrout, Concerning cellular decompositions of 3manifolds that yield 3manifolds, Trans. Amer. Math. Soc. 133 (1968), 307332. MR 37 #5859. MR 0230296 (37:5859)
 [5]
M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 7476. MR 22 #8470b. MR 0117695 (22:8470b)
 [6]
A. V. Černavskii, Local contractibility of the homeomorphism group of a manifold, Dokl. Akad. Nauk SSSR 182 (1968), 510513 = Soviet Math. Dokl. 9 (1968), 11711174. MR 38 #5241. MR 0236948 (38:5241)
 [7]
R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 6388. MR 44 #1032. MR 0283802 (44:1032)
 [8]
S. Eilenberg and R. L. Wilder, Uniform local connectedness and contractibility, Amer. J. Math. 64 (1942), 613622. MR 4, 87. MR 0007100 (4:87e)
 [9]
R. Geoghegan, On spaces of homeomorphisms, embeddings, and functions. I, Topology 11 (1972), 159177. MR 45 #4349. MR 0295281 (45:4349)
 [10]
R. Geoghegan and D. W. Henderson, Stable function spaces (to appear). MR 0339269 (49:4029)
 [11]
O. Hanner, Some theorems on absolute neighborhood retracts, Ark. Mat. 1 (1951), 389408. MR 13, 266. MR 0043459 (13:266d)
 [12]
W. Haver, Homeomorphisms and maps, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N. Y., 1970), State Univ. of New York at Binghamton, Binghamton, N. Y., 1971, pp. 112121. MR 44 #1000. MR 0283770 (44:1000)
 [13]
, Monotone mappings of a twodisk onto itself which fix the disk's boundary can be canonically approximated by homeomorphisms, Pacific J. Math. (to appear).
 [14]
W. Hurewicz and H. Wallman, Dimension theory, Princeton Math. Series, vol. 4, Princeton Univ. Press, Princeton, N. J., 1941. MR 3, 312. MR 0006493 (3:312b)
 [15]
C. Lacher, Celllike mappings. I, Pacific J. Math. 30 (1969), 717731. MR 40 #4941. MR 0251714 (40:4941)
 [16]
, Celllike mappings. II, Pacific J. Math. 35 (1970), 649660. MR 43 #6936. MR 0281217 (43:6936)
 [17]
J. A. Lees, Immersions and surgeries of topological manifolds, Bull. Amer. Math. Soc. 75 (1969), 529534. MR 39 #959. MR 0239602 (39:959)
 [18]
W. K. Mason, The space of homeomorphisms of a compact manifold onto itself is homeomorphic to minus any 6compact set, Amer. J. Math. 92 (1970), 541551. MR 43 #6953. MR 0281234 (43:6953)
 [19]
, The space of all selfhomeomorphisms of a twocell which fix the cell's boundary in an absolute retract, Trans. Amer. Math. Soc. 161 (1971), 185205. MR 44 #3283. MR 0286067 (44:3283)
 [20]
L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271294. MR 45 #4431. MR 0295365 (45:4431)
 [1]
 J. W. Alexander, On the deformation of an ncell, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 406407.
 [2]
 R. D. Anderson, Strongly negligible sets in Fréchet manifolds, Bull. Amer. Math. Soc. 75 (1969), 6467. MR 38 #6634. MR 0238358 (38:6634)
 [3]
 , Spaces of homeomorphisms of finite graphs (to appear).
 [4]
 S. Armentrout, Concerning cellular decompositions of 3manifolds that yield 3manifolds, Trans. Amer. Math. Soc. 133 (1968), 307332. MR 37 #5859. MR 0230296 (37:5859)
 [5]
 M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 7476. MR 22 #8470b. MR 0117695 (22:8470b)
 [6]
 A. V. Černavskii, Local contractibility of the homeomorphism group of a manifold, Dokl. Akad. Nauk SSSR 182 (1968), 510513 = Soviet Math. Dokl. 9 (1968), 11711174. MR 38 #5241. MR 0236948 (38:5241)
 [7]
 R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 6388. MR 44 #1032. MR 0283802 (44:1032)
 [8]
 S. Eilenberg and R. L. Wilder, Uniform local connectedness and contractibility, Amer. J. Math. 64 (1942), 613622. MR 4, 87. MR 0007100 (4:87e)
 [9]
 R. Geoghegan, On spaces of homeomorphisms, embeddings, and functions. I, Topology 11 (1972), 159177. MR 45 #4349. MR 0295281 (45:4349)
 [10]
 R. Geoghegan and D. W. Henderson, Stable function spaces (to appear). MR 0339269 (49:4029)
 [11]
 O. Hanner, Some theorems on absolute neighborhood retracts, Ark. Mat. 1 (1951), 389408. MR 13, 266. MR 0043459 (13:266d)
 [12]
 W. Haver, Homeomorphisms and maps, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N. Y., 1970), State Univ. of New York at Binghamton, Binghamton, N. Y., 1971, pp. 112121. MR 44 #1000. MR 0283770 (44:1000)
 [13]
 , Monotone mappings of a twodisk onto itself which fix the disk's boundary can be canonically approximated by homeomorphisms, Pacific J. Math. (to appear).
 [14]
 W. Hurewicz and H. Wallman, Dimension theory, Princeton Math. Series, vol. 4, Princeton Univ. Press, Princeton, N. J., 1941. MR 3, 312. MR 0006493 (3:312b)
 [15]
 C. Lacher, Celllike mappings. I, Pacific J. Math. 30 (1969), 717731. MR 40 #4941. MR 0251714 (40:4941)
 [16]
 , Celllike mappings. II, Pacific J. Math. 35 (1970), 649660. MR 43 #6936. MR 0281217 (43:6936)
 [17]
 J. A. Lees, Immersions and surgeries of topological manifolds, Bull. Amer. Math. Soc. 75 (1969), 529534. MR 39 #959. MR 0239602 (39:959)
 [18]
 W. K. Mason, The space of homeomorphisms of a compact manifold onto itself is homeomorphic to minus any 6compact set, Amer. J. Math. 92 (1970), 541551. MR 43 #6953. MR 0281234 (43:6953)
 [19]
 , The space of all selfhomeomorphisms of a twocell which fix the cell's boundary in an absolute retract, Trans. Amer. Math. Soc. 161 (1971), 185205. MR 44 #3283. MR 0286067 (44:3283)
 [20]
 L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271294. MR 45 #4431. MR 0295365 (45:4431)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
57E05,
57A20
Retrieve articles in all journals
with MSC:
57E05,
57A20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403623750
PII:
S 00029947(1974)03623750
Keywords:
Spaces of homeomorphisms,
cellular mappings,
closure of the space of homeomorphisms,
compact manifolds,
infinite dimensional manifolds,
homogeneous spaces
Article copyright:
© Copyright 1974
American Mathematical Society
