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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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: For any mapping $ f:V \to M$ (not necessarily an embedding), where V and M are differentiable manifolds without boundary of dimensions k and n, respectively, V compact, let $ {[V \subset M]_f} = {\pi _1}({M^V},E,f)$, 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 $ {[V \subset M]_f}$. For example, if $ k \geqslant 3,n = 2k$, and M is simply connected, $ {[{S^k} \subset M]_f}$ corresponds to $ {\pi _2}M$ or $ {\pi _2}M \otimes {Z_2}$, depending on whether k is odd or even. In the metastable range, i.e., $ 3(k + 1) > 2n$, a natural Abelian affine structure on $ {[V \subset M]_f}$ is defined: if, further, f is an embedding $ {[V \subset M]_f}$ 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 $ {\pi _1}({M^V},f)$ on $ {[V \subset M]_f}$.

A spectral sequence is constructed converging to a theory $ {H^\ast}(f)$. If $ 3(k + 1) < 2n, {H^0}(f) \cong {[V \subset M]_f}$ provided the latter is nonempty. A single obstruction $ \Gamma (f) \in {H^1}(f)$ is also defined, which must be zero if f is homotopic to an embedding; this condition is also sufficient if $ 3(k + 1) \leqslant 2n$. The $ {E_2}$ terms are cohomology groups of the reduced deleted product of V with coefficients in sheaves which are not even locally trivial. $ {[{S^k} \subset M]_f}$ is specifically computed in terms of generators and relations if $ n = 2k, k \geqslant 3$ (Theorem 6.0.2).


References [Enhancements On Off] (What's this?)

  • [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

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