Isotopy groups

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).