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Elimination of certain Thom-Boardman singularities of order two

1982; Mathematical Society of Japan; Volume: 34; Issue: 2 Linguagem: Inglês

10.2969/jmsj/03420241

1881-1167

Yoshifumi Ando,

Mathematical functions and polynomials

In this paper we will study the problem of deforming a differentiable map of a differentiable manifold $N$ into a differentiable manifold $P$ in the homotopy class to a differentiable maP which does not admit particularly complicated Thom-Boardman singularities of order two.Let $\Sigma^{I}(N, P)$ denote the Thom-Boardman singularity with symbol $I$ which is defined in the jet-space $J^{r}(N, P)$ where $I$ denote either (i) for $r=1$ , or $(i, j)$ $\iota$ for $r=2$ ([2], [11] and [18]).Let $\Sigma^{I}^{-}(N, P)$ denote the closure of $\Sigma^{I}(N, P)$ in $J^{r}(N, P)$ , and $\nu_{I}$ the codimension of $\Sigma^{I}(N, P)$ in $J^{r}(N, P)$ .Let $N$ be a closed differentiable manifold, $n=\dim N$ and $p=\dim P$ .The canonical fiber of $\Sigma^{I}(N, P)$ over $N\times P$ will be denoted by $\Sigma^{I}(n, p)$ .We will define in \S 2 and \S 6 the dual class $[\Sigma^{I}-(N, P)]$ in $H^{\nu_{I}}(N\times P;G)$ of the Thom-Boardman singularity $\Sigma^{I}(N, P)$ .The coefficient group $G$ denotes either $Z$ or $Z_{2}$ depending on whether $\Sigma^{I}(n,$ $ p\rangle$ is orientable or not.When $G$ is $Z$ , we assume $N$ and $P$ to be orientable mani- folds.For a differentiable map $f:N\rightarrow P$ we denote the class $(id_{N}\times f)^{*}([\Sigma^{I}-(N, P)])$ in $H^{\nu_{I}}(N;G)$ by $c^{I}(TN, f^{*}(TP))$ .We will give in \S 5 a formula to calculate the dual class $c^{I}(TN, f^{*}(TP))$ in a finite process in terms of the characteristic classes of $N$ and $P$ .The $Z_{2}$ -reduction of these dual classes coincides with those which have been defined in [15] under the sheaf homology (cohomology resp.\ranglegroups with closed supports.We will use the singular homology (cohomology resp.)groups in our definition.We will show the following two applications of the dual classes.Let $\Omega^{I}(N, P)$ denote the union of all Thom-Boardman singularities with symbol smaller than or equal to $I$ in the lexicographic order.Let $C_{\Omega^{I}}^{\infty}(N, P)$ denote the space of all differentiable maps, $f:N\rightarrow P$ such that the image of $j^{r}f:N\rightarrow J^{r}(N, P)$ is contained in $\Omega^{I}(N, P)1$ with $C^{\infty}$ -topology.Let $\Gamma_{\Omega^{I}}(N)$ denote the space of all continuous sections of the fiber bundle of $J^{r}(N, P)$ over $N$ such that the image of a section is contained in $\Omega^{I}(N, P)$ equipped with compact- open topology.Let $\Omega(N, P)$ denote $\Omega^{I}(N, P)\backslash \Sigma^{I}(N, P)$ and we consider similarly 242 Y. ANDO $C_{\Omega}^{\infty}(N, P)$ and $\Gamma_{\Omega}(N)$ .Then we have a canonical cotinuous map $j^{r}$ : $ C_{\Omega}^{\infty}(N, P)\rightarrow$ $\Gamma_{\Omega}(N)$ .We will say that $\Omega(N, P)$ is $\pi_{0}$ -integrable when the induced map $(j^{r})^{*}:$ $\pi_{0}(C_{\Omega}^{\infty}(N, P))\rightarrow\pi_{0}(\Gamma_{\Omega}(N))$ is surjective and that a symbol $I$ is good when every fibre of $\Omega^{I}(N, P)$ and of $\Omega(N, P)$ over $N\times P$ are connected and simply connected.The $\pi_{0}$ -integrability has been discussed in [4] and [14] extensively and also in [1].Note that $I$ is good if $co\dim\Sigma^{K}>2$ for every $K$ with $K\geqq I$ , and hence $I$ is good in most case.The following theorem will be proved in \S 6.THEOREM 1.Let $N,$ $P$ and $f$ be as above.Let I be any good symbol such that $co\dim\Sigma^{I}(N, P)=n$ except for the case that $r=2,$ $p-n+i=1$ and $i>j$ .Let $\Omega(N, P)$ be $\pi_{0}$ -integrable.Then there exists a differentiable map $g$ of $C\partial(N, P)$ such that $]^{\gamma}g$ is homotopic to $]^{\gamma}f$ as maps of $N$ into $\Omega^{I}(N, P)$ (hence $g$ is homo- tolnc to f) if and only if the dual class $c^{I}(TN, f^{*}(TP))$ vanishes.In \S 7 we will treat continuous maps which are not homotopic to any $C^{\infty}$ -stable map.J. N. Mather has proved in [10] that $C^{\infty}$ -stable maps are dense in the space of all proper differentiable maps of $C^{\infty}(N, P)$ in the 'nice range'.We will obtain the following result outside of the nice range.THEOREM 2. Let $f$ be a continuous map of a closed differentiable manifold $N$ into a differentiable manifold P. We assume that there exists an integer $i$ such that(1) $(p-n+i)\{i+\frac{1}{2}i(i+1)\}-i^{2}-(p-n+\iota)^{2}+1>n$ , and(2) either (i) the determinant of the following $(p-n+i)$ -matrix whose $(s, t)$ component is the Stiefel-Whitney class $W_{i+s-t}(\gamma)$ of $\gamma=TN-f^{*}(TP)$ $(W_{i}(\gamma)..\cdot.\cdot.W_{i-1}(.\gamma)W_{i+1}(\gamma).\dot{W}_{i+1}(\gamma)W_{i}(\gamma)\dot{W}_{i-1}(\gamma)]$ is not zero, $or$ (ii) both of $n-P$ and $i$ are even, both of $N$ and $P$ are orientable and the determinant of the following $(p-n+i)/2$ -matrix whose $(s, t)$ component is ihe Pontrjagin class $P_{(i/2)+S-J}(\gamma)$ $[]-$Thom-Boardman singularities 243 is not an element of 2-torsion.Then $f$ is not homotopic to any $C^{\infty}$ -stable map (especially when $P$ is $R^{p}$ , there exists no $C^{\infty}$ -stable map in $C^{\infty}(N, R^{p})$ .This theorem follows from more general Theorem 7.4.In \S 7 we will give examples of manifolds $N$ and $P$ such that there exists no $C^{\infty}$ -stable map in $C^{\infty}(N, P)$ .All manifolds, fiber bundles and maps will be differentiable of class $C^{\infty}$ unless otherwise stated.All manifolds will be paracompact and Hausdorff.\S 1. Notations and preliminaries.Let $\xi$ and $\eta$ denote respectively differentiable real vector bundles, $E\rightarrow X$ and $F\rightarrow X$ of dimensions $n$ and $P$ over a manifold $X$ .Let $E_{x}$ and $F_{x}$ be respectively fibres of $E$ and $F$ over a point $x$ in $X$ .Let Hom $(\xi, \eta)$ denote the union of all linear maps of $E_{x}$ into $F_{x},\bigcup_{x\in X}$ Hom $(E_{x}, F_{x})$ , which becomes naturally a real vector bundle over $X$ .We begin with recalling the definition of Thom-Boardman singularities (see [2], [11] and especially [15]).For convenience we put $J^{1}(\xi, \eta)=Hom(\xi, \eta)$ and $J^{2}(\xi, \eta)=Hom(\xi, \eta)\oplus Hom(\xi\circ\xi, \eta)$ where $\xi\circ\xi$ denote the symmetric product of $\xi$ .In the sequel we will denote an element of $J^{2}(\xi, \eta)$ over a point $x$ in $X$ by \ l a n g l e$ \alpha,$ $\beta$ ) where $\alpha$ (resp.$\beta$ ) is an element of Hom $(E_{x}, F_{x})$ (resp.Hom $(E_{x}\circ E_{x},$ $F_{x})$ ).An element $(\alpha, \beta)$ of $J^{2}(E_{x}, F_{x})$ determines a linear map $\tilde{\beta}$ : Ker $(\alpha)\rightarrow Hom(Ker(\alpha)$ , Cok $(\alpha))$ which is induced from the projection of $F_{x}$ onto Cok $(\alpha)$ and the iso- morphism of Hom $(E_{x}\otimes E_{x}, F_{x})$ onto Hom ( $E_{x}$ , Hom $(E_{x},$ $F_{x})$ ).DEFINITION 1.1.Let $\Sigma^{i}(\xi, \eta)$ denote the space of all elements of $J^{1}(\xi, \eta)$ such that the dimension of Ker $(\alpha)$ is $i$ .Let $\Sigma^{i,j}(\xi, \eta)$ denote the space of all elements $(\alpha, \beta)$ of $J^{2}(\xi, \eta)$ such that $\alpha\in\Sigma^{i}(\xi, \eta)$ and that the dimension of Ker $(\tilde{\beta})$ is $j$ .In the sequel $I$ means either (i) or $(i, j)$ .$\Sigma^{i,j}(\xi, \eta)$ is nonempty if and only if (i) $n\geqq i\geqq j\geqq 0$ (ii) $i\geqq n-p$ and (iii) $i=j$ for $i=n-p$ .We call $\Sigma^{I}(\xi, \eta)$ the Thom-Boardman singularity with symbol $I$ .When $\xi$ and $\eta$ are trivial bundles $R^{n}$ and $R^{p}$ over a single point, we simply write $\Sigma^{I}(n, p)$ (resp.$J^{r}(n,$ $p)$ ) in place of $\Sigma^{I}(\xi, \eta)$ (resp.$J^{r}(\xi,$ $\eta)$ ).It is clear that $\Sigma^{I}(\xi, \eta)=\bigcup_{x\in X}\Sigma^{I}(E_{x}, F_{x})$ .It is shown in [2], [11] and [15] that $\Sigma^{I}(\xi, \eta)$ is a regular submanifold of $J^{r}(\xi, \eta)$ .The codimension of $\Sigma^{i}(\xi, \eta)$ in $J^{1}(\xi, \eta)$ is $i(p-n+i)$ and the codimension of $\Sigma^{i,j}(\xi, \eta)$ in $J^{2}(\xi, \eta),$ $(i+i\circ j)(p-n+i)-j(i-j)$ where an integer $i\circ j$ denotes the dimen- sion $j(i-j)+(i/2)j(j+1)$ of $R^{i}\circ R^{j}$ .If we provide $\xi$ and $\eta$ with metrics, then we can provide $J^{r}(\xi, \eta)$ with a metric.Let $S^{r}(\xi, \eta)$ denote the associated sphere bundle of $J^{r}(\xi, \eta)$ .We put $\Sigma_{0}^{I}(\xi, \eta)=\Sigma I(\xi, \eta)\cap S^{r}(\xi, \eta)$ .Then $\Sigma_{0}^{I}(\xi, \eta)$ is empty for $I=(n)$ and $I=(n, n)$ .