Isotopy groups

Author:
Lawrence L. Larmore

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

MSC:
Primary 57R40

MathSciNet review:
487040

Full-text PDF Free Access

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]**James C. Becker,*Cohomology and the classification of liftings*, Trans. Amer. Math. Soc.**133**(1968), 447–475. MR**0236924**, 10.1090/S0002-9947-1968-0236924-4**[2]**Jean-Pierre Dax,*Étude homotopique des espaces de plongements*, Ann. Sci. École Norm. Sup. (4)**5**(1972), 303–377 (French). MR**0321110****[3]**André Haefliger,*Plongements différentiables dans le domaine stable*, Comment. Math. Helv.**37**(1962/1963), 155–176 (French). MR**0157391****[4]**André Haefliger,*Points multiples d’une application et produit cyclique réduit*, Amer. J. Math.**83**(1961), 57–70 (French). MR**0120655****[5]**I. M. Hall,*The generalized Whitney sum*, Quart. J. Math. Oxford Ser. (2)**16**(1965), 360–384. MR**0187245****[6]**Sze-tsen Hu,*Homotopy theory*, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR**0106454****[7]**Lawrence L. Larmore,*Obstructions to embedding and isotopy in the metastable range*, Rocky Mountain J. Math.**3**(1973), 355–375. MR**0356088****[8]**Lawrence L. Larmore,*Twisted cohomology theories and the single obstruction to lifting*, Pacific J. Math.**41**(1972), 755–769. MR**0353315****[9]**L. L. Larmore and E. Thomas,*Group extensions and principal fibrations*, Math. Scand.**30**(1972), 227–248. MR**0328935****[10]**L. L. Larmore and E. Thomas,*Group extensions and twisted cohomology theories*, Illinois J. Math.**17**(1973), 397–410. MR**0336744****[11]**Mark Mahowald and Robert Rigdon,*Obstruction theory with coefficients in a spectrum*, Trans. Amer. Math. Soc.**204**(1975), 365–384. MR**0488058**, 10.1090/S0002-9947-1975-0488058-5**[12]**Hans A. Salomonsen,*On the existence and classification of differentiable embeddings in the metastable range*(to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57R40

Retrieve articles in all journals with MSC: 57R40

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