An unknotting theorem in -manifolds

Author:
Vo Thanh Liem

Journal:
Proc. Amer. Math. Soc. **82** (1981), 125-132

MSC:
Primary 57N20; Secondary 57N37

DOI:
https://doi.org/10.1090/S0002-9939-1981-0603615-9

MathSciNet review:
603615

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Abstract: In this note, we prove the following unknotting theorem.

Theorem. Let *be a* *-manifold and let* be a homotopy such that *and* *are* *-deficient embeddings. Then, there is an isotopy* *such that* *and* . *Moreover, if* is limited by an open cover *of* *and is stationary on a closed subset* *of* , *then we may choose* *to also be limited by* *and to be the identity on* .

However, a similar unknotting theorem for -embeddings does not hold true in and .

**[1]**T. A. Chapman,*Lectures on Hilbert cube manifolds*, American Mathematical Society, Providence, R. I., 1976. Expository lectures from the CBMS Regional Conference held at Guilford College, October 11-15, 1975; Regional Conference Series in Mathematics, No. 28. MR**0423357****[2]**Ross Geoghegan (ed.),*Open problems in infinite-dimensional topology*, The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979, pp. 287–338 (1980). MR**583711****[3]**Richard E. Heisey,*Manifolds modelled on the direct limit of Hilbert cubes*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 609–619. MR**537754****[4]**R. E. Heisey and H. Torunzcyk,*On the topology of direct limits of*ANR'*s*(preprint).**[5]**Vo Thanh Liem,*An 𝛼-approximation theorem for 𝑄^{∞}-manifolds*, Topology Appl.**12**(1981), no. 3, 289–304. MR**623737**, https://doi.org/10.1016/0166-8641(81)90007-9**[6]**Steve Ferry,*The homeomorphism group of a compact Hilbert cube manifold is an 𝐴𝑁𝑅*, Ann. of Math. (2)**106**(1977), no. 1, 101–119. MR**0461536**, https://doi.org/10.2307/1971161

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1981-0603615-9

Keywords:
Hilbert cube,
direct limit space,
-set,
inductive -set,
-deficient,
isotopy,
unknotting theorem

Article copyright:
© Copyright 1981
American Mathematical Society