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Isotopy Groups

Lawrence L. Larmore
Transactions of the American Mathematical Society
Vol. 239 (May, 1978), pp. 67-97
DOI: 10.2307/1997848
Stable URL: http://www.jstor.org/stable/1997848
Page Count: 31
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Isotopy Groups
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Abstract

For any mapping $f: V \rightarrow M$ (not necessarily an embedding), where $V$ and $M$ are differentiable manifolds without boundary of dimensions $k$ and $n$,respectively, $V$ compact, let $\lbrack V \subset M \rbrack_f = \pi_1(M^v, E, f)$ 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 $\lbrack V \subset M \rbrack_f$. For example, if $k \geqslant 3, n = 2k$, and $M$ is simply connected, $\lbrack S^k \subset M \rbrack_f$ corresponds to $\pi_2M$ or $\pi_2M \otimes Z_2$, depending on whether $k$ is odd or even. In the metastable range, i.e., $3(k + 1) > 2n$, a natural Abelian affine structure on $\lbrack V \subset M \rbrack_f$ is defined: if, further, $f$ is an embedding $\lbrack V \subset M \rbrack_f$ is then an Abelian group. The set of istopy classes of embeddings homotopic to $f$ is the set of orbits of the obvious left action of $\pi_1(M^V, f)$ on $\lbrack V \subset M \rbrack_f$. A spectral sequence is constructed coverging to a theory $H^\ast(f)$. If $3(k + 1) < 2n, H^0(f) \cong \lbrack V \subset M \rbrack_f$ 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. $\lbrack S^k \subset M \rbrack_f$ is specifically computed in terms of generators and relations if $n = 2k, k \geqslant 3$ (Theorem 6.0.2).

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