Enumerating embeddings of n-manifolds in Euclidean (2n-1)-space
Tsutomu Yasui · Journal of the Mathematical Society of Japan · 1984
Throughout this paper, "n-manifold" and "embedding" will mean closed connected differentiable manifold of dimension $n$ and differentiable embedding, respectively.Let $[M\subset R^{m}]$ denote the set of isotopy classes of embeddings of a manifold $M$ into Euclidean m-space.It is known that for an n-manifold $M$ , (1) (Whitney [24]) the set $[M\subset R^{2n+2}]$ consists of only one element if $n\geqq 1$ , (2) (Wu [25]) the set $[M\subset R^{2n+1}]$ consists of only one element if $n\geqq 2$ , (3) (Haefliger [6], Bausum [1], Rigdon [13] etc.) if $n\geqq 4$ , then, as a set, $[M\subset R^{2n}]=\{\begin{array}{ll}H^{n-1}(M;Z) for n\equiv 1(2), w_{1}(M)=0,H^{n-1}(M;Z_{2}) for n\equiv 1(2), w_{1}(M) eq 0,or n\equiv 0(2), w_{1}(M)=0, Z\cross\rho_{2}H^{n-1}(M;Z) for n\equiv 0(2), w_{1}(M) eq 0.\end{array}$ The purpose of this paper is to inquire into the question of whether or not the set $[M\subset R^{2n-1}]$ for an n-manifold $M$ , if it is not empty, can be described in terms of the cohomology of $M$ , its characteristic classes and the cohomology operations.We shall study $[M\subset R^{2n-1}]$ along the lines of Haefliger [5], [6].Let $X^{2}$ be the product $X\cross X$ of $X$ and let $\Delta X$ be the diagonal in $X^{2}$ .The cyclic group of order 2, $Z_{2}$ , acts on $X^{2}$ via the map $t:X^{2}arrow X^{2}$ defined by $t(x, y)$ $=(y, x)$ , where $\Delta X$ is the fixed point set of this action.The quotient space $X^{*}=(X^{2}-\Delta X)/Z_{2}$ is called the reduced symmetric product of $X$ .Let $P^{m}$ denote the real projective space of dimension $m(m\leqq\infty)$ and let $\xi:x*arrow P^{\infty}$ denote the classifying map of the double covering $X^{2}-\Delta Xarrow X^{*}$ .Then the first Stiefel-Whitney class of this double covering is given by