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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
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).

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Keywords: Isotopy, obstructions to embedding, twisted cohomology
Article copyright: © Copyright 1978 American Mathematical Society