Isotopy groups

Author:
Lawrence L. Larmore

Journal:
Trans. Amer. Math. Soc. **239** (1978), 67-97

MSC:
Primary 57R40

DOI:
https://doi.org/10.1090/S0002-9947-1978-0487040-4

MathSciNet review:
487040

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Abstract | References | Similar Articles | Additional Information

Abstract: For any mapping (not necessarily an embedding), where *V* and *M* are differentiable manifolds without boundary of dimensions *k* and *n*, respectively, *V* compact, let , i.e., the set of isotopy classes of embeddings with a specific homotopy to *f* (*E* = space of embeddings). The purpose of this paper is to enumerate . For example, if , and *M* is simply connected, corresponds to or , depending on whether *k* is odd or even. In the metastable range, i.e., , a natural Abelian affine structure on is defined: if, further, *f* is an embedding is then an Abelian group. The set of isotopy classes of embeddings homotopic to *f* is the set of orbits of the obvious left action of on .

A spectral sequence is constructed converging to a theory . If provided the latter is nonempty. A single obstruction is also defined, which must be zero if *f* is homotopic to an embedding; this condition is also sufficient if . The terms are cohomology groups of the reduced deleted product of *V* with coefficients in sheaves which are not even locally trivial. is specifically computed in terms of generators and relations if (Theorem 6.0.2).

**[1]**J. C. Becker,*Cohomology and the classification of liftings*, Trans. Amer. Math. Soc.**133**(1968), 447-475. MR**38**#5217. MR**0236924 (38:5217)****[2]**Jean-Pierre Dax,*Etude homotopique des espaces de plongements*, Ann. Sci. École Norm. Sup. (4)**5**(1972), 303-377. MR**47**#9643. MR**0321110 (47:9643)****[3]**A. Haefliger,*Plongements différentiates dans le domaine stable*, Comment. Math. Helv.**37**(1962/63), 155-176. MR**28**#625. MR**0157391 (28:625)****[4]**-,*Points multiples d'une application et produit cyclique réduit*, Amer. J. Math.**83**(1961), 57-70. MR**22**# 11404. MR**0120655 (22:11404)****[5]**I. M. Hall,*The generalized Whitney sum*, Quart. J. Math. Oxford Ser. (2)**16**(1965), 360-384. MR**32**#4698. MR**0187245 (32:4698)****[6]**S.-T. Hu,*Homotopy theory*, Academic Press, New York and London, 1959. MR**21**#5186. MR**0106454 (21:5186)****[7]**L. L. Larmore,*Obstructions to embedding and isotopy in the metastable range*, Rocky Mountain J. Math.**3**(1973), 355-375. MR**50**#8559. MR**0356088 (50:8559)****[8]**-,*Twisted cohomology theories and the single obstruction to lifting*, Pacific J. Math.**41**(1972), 755-769. MR**50**#5799. MR**0353315 (50:5799)****[9]**L. L. Larmore and E. Thomas,*Group extensions and principal fibrations*, Math. Scand.**30**(1972), 227-248. MR**48**#7277. MR**0328935 (48:7277)****[10]**-,*Group extensions and twisted cohomology theories*, Illinois J. Math.**17**(1973), 397-410. MR**49**#1517. MR**0336744 (49:1517)****[11]**M. Mahowald and R. Rigdon,*Obstruction theory with coefficients in a spectrum*, Trans. Amer. Math. Soc.**204**(1975), 365-384. MR**0488058 (58:7630)****[12]**Hans A. Salomonsen,*On the existence and classification of differentiable embeddings in the metastable range*(to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0487040-4

Keywords:
Isotopy,
obstructions to embedding,
twisted cohomology

Article copyright:
© Copyright 1978
American Mathematical Society