Smooth maps with singularities of bounded K-codimensions

TL;DR

This paper establishes the relative homotopy principle for smooth maps with controlled singularities and explores the structure of homotopy self-equivalences of manifolds based on singularity codimensions.

Contribution

It proves the relative homotopy principle for maps with specific singularities and analyzes the filtration of self-equivalence groups by singularity codimensions.

Findings

Proved the relative homotopy principle under mild conditions.

Analyzed the filtration of the group of homotopy self-equivalences.

Connected singularity types with manifold self-equivalence structures.

Abstract

We will prove the relative homotopy principle for smooth maps with singularities of a given {\cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {\cal K}-codimensions.

Equations

$${\mathfrak{h}}_0(P)\subset {\mathfrak{h}}_1(P)\subset \cdots \subset {\mathfrak{h}}_{\ell }(P)\subset \cdots \subset \mathfrak{h}(P).$$

$${\mathfrak{h}}_0(P){\mathfrak{h}}_{{\ell }_1}(P){\mathfrak{h}}_{{\ell }_2}(P)\cdots {\mathfrak{h}}_{{\ell }_d}(P)\mathfrak{h}(P).$$

$$\mathrm{codim}{\Sigma }^I(n,p)\geqq (p-n+i_1)i_1+(1/2){\Sigma }_{j=2}^{\ell }{i}_j(i_j+1).$$

$$tf:\theta (N{)}_x\to \theta (f{)}_x$$

$$T_z(J_{x,y}^k(N,P))⟶{\mathfrak{m}}_x\theta (f{)}_x/{\mathfrak{m}}_x^{k+1}\theta (f{)}_x$$

$$\dim {\mathfrak{m}}_x\theta (f{)}_x/(tf({\mathfrak{m}}_x\theta (N{)}_x)+f^{\ast }({\mathfrak{m}}_y)(\theta (f{)}_x)),$$

$$\mathbf{d}$$

$${\mathbf{d}}_1$$

$${\mathbf{d}}_z(w,\mathbf{v})=(w,d(j^{k-1}f)(\mathbf{v}))\text{ and }({\mathbf{d}}_1{)}_z(w,\mathbf{v})=(w,df(\mathbf{v})).$$

$$\mathbf{K}({\Sigma }^{I^{\prime }}{)}_w=(w,\mathrm{Ker}(d_{x}f)).$$

$$\mathbf{d}(\mathbf{K}({\Sigma }^{I^{\prime }}{)}_w)\cap ({\pi }_{k-1}^k|{\Sigma }^{I^{\prime }}(N,P){)}^{\ast }(T({\Sigma }^I(N,P){)}_w=\{0\}$$

$$\mathbf{d}(\mathbf{K}(V^i{)}_z)\cap ({\pi }_{k-1}^k|V^i(N,P){)}^{\ast }(T(V^i(N,P{)}^{(k-1)}{)}_z=\{0\}.$$

$$V_I(n,p)=\mathcal{O}(n,p)\cap {\Sigma }^I(n,p).$$

$$\text{codim}{\Sigma }^J(n,p)\geqq |n-p|+k-1\geqq n+1.$$

$$V_{J^{\ast }}(n,p{)}^{(k-1)}={\pi }_{k-1}^k(\mathcal{O}(n,p))\cap {\Sigma }^{J\ast }(n,p).$$

$${\Sigma }^J(n,p)=({\pi }_{k-1}^k{)}^{-1}({\Sigma }^{J^{\ast }}(n,p))\text{ and }\mathcal{O}(n,p)\subset ({\pi }_{k-1}^k{)}^{-1}({\pi }_{k-1}^k(\mathcal{O}(n,p))).$$

$$({\pi }_k^{\infty }{)}^{-1}({\mathcal{O}}_{\ell }^k(n,p))={\mathcal{O}}_{\ell }^{\infty }(n,p).$$

$$(p-n+i)(\frac{1}{2}i(i+1)-p+n)-i^2\geqq n,$$

$$(p-n+i)(\frac{1}{2}i(i+1)-p+n)-i^2\geqq n+\ell ,$$

$${\Sigma }^i(n+\ell ,p+\ell )\subset \Sigma (n+\ell ,p+\ell ;k).$$

$$0\to \mathrm{Ker}(\alpha )\to L\oplus L^{\perp }\oplus \mathrm{Ker}(\alpha )\stackrel{\alpha }{\to }M\oplus M^{\perp }\oplus \mathrm{Im}(\alpha {)}^{\perp }\to \mathrm{Im}(\alpha {)}^{\perp }\to 0$$

$$(u_1,\cdots ,u_m)\text{, }(u_{m+1},\cdots ,u_{m+n-i})\text{ and }(u_{m+n-i+1},\cdots ,u_{m+n})$$

$$(y_{1},\cdots,y_{m})\text{, }(y_{m+1},\cdots,y_{m+n-i})\text{ and }(y_{m+n-i+1},\cdots,y_{m+p})\$$

$$x_j=x_j(u_1,\cdots ,u_{m+n})(1\le j\le m)\text{ and }{x}_j=u_j(m+1\le j\le m+n)$$

$$y_j\circ f(x_1,\cdots ,x_{m+n})=x_j(1\le j\le m).$$

$$y_j\circ g(\stackrel{\bullet }{x})=y_j\circ f(0,\cdots ,0,\stackrel{\bullet }{x})(m+1\le j\le m+p).$$

$$\pi \left(\sum _{j=1}^m{a}_{j}tf(\frac{\partial }{\partial xj})+\sum _{j=m+1}^{m+p}{a}_j(\frac{\partial }{\partial yj}\circ f)\right)=\sum _{j=m+1}^{m+p}{a}_j^{\prime }(\frac{\partial }{\partial yj}\circ g),$$

$$\begin{array}[c]{l}tf(\partial/\partial x_{j})=(\partial/\partial y_{j})\circ f+\sum_{t=m+1}^{m+p}(\partial y_{t}\circ f/\partial x_{j})(\partial/\partial y_{t})\circ f\text{ \ }(1\leq j\leq m),\\ tf(\partial/\partial x_{j})=\sum_{t=m+1}^{m+p}(\partial y_{t}\circ f/\partial x_{j})(\partial/\partial y_{t})\circ f\text{ \ }(m+1\leq j\leq m+n),\\ (\partial y_{t}\circ f/\partial x_{j})(0,\cdots,0,\overset{\bullet}{x})=(\partial y_{t}\circ g/\partial x_{j})(\overset{\bullet}{x})\text{ \ }(m+1\leq t\leq m+p).\end{array}$$

$$y_t\circ f(x_1,\cdots ,x_{m+n})-y_t\circ f(0,\cdots ,0,\stackrel{\bullet }{x})=\sum _{u=1}^m{x}_u{b}_u(x_1,\cdots ,x_{m+n}),$$

$$\partial y_t\circ f/\partial x_j-\partial y_t\circ g/\partial x_j=\sum _{u=1}^m{x}_u(\partial b_u/\partial x_j)(m+1\le j\le m+n).$$

$$\left|\begin{array}[c]{ccc}W_{i}&\cdots&W_{n-p+1}\\ \vdots&\ddots&\vdots\\ W_{p-n+2i-1}&\cdots&W_{i}\end{array}\right|\text{ \ \ and \ \ }\left|\begin{array}[c]{ccc}P_{v}&\cdots&P_{u+1}\\ \vdots&\ddots&\vdots\\ P_{2v-u-1}&\cdots&P_{v}\end{array}\right|$$

Full text

1 Introduction

Let $N$ and $P$ be smooth ($C^{\infty}$) manifolds of dimensions $n$ and $p$ respectively. Let $J^{k}(N,P)$ denote the $k$-jet space of the manifolds $N$ and $P$ with the projections $\pi_{N}^{k}$ and $\pi_{P}^{k}$ onto $N$ and $P$ mapping a jet onto its source and target respectively. The canonical fiber is the $k$-jet space $J^{k}(n,p)$ of $C^{\infty}$-map germs $(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)$. Let $\mathcal{K}$ denote the contact group defined in [MaIII]. Let $\mathcal{O}(n,p)$ denote a $\mathcal{K}$-invariant nonempty open subset of $J^{k}(n,p)$ and let $\mathcal{O}(N,P)$ denote an open subbundle of $J^{k}(N,P)$ associated to $\mathcal{O}(n,p)$. In this paper a smooth map $f:N\rightarrow P$ is called an $\mathcal{O}$-regular map if $j^{k}f(N)\subset\mathcal{O}(N,P)$.

We will study what is called the homotopy principle for $\mathcal{O}$-regular maps. As for the long history of the several types of homotopy principles and their applications we refer to the Smale-Hirsch Immersion Theorem ([Sm] and [H]), the Feit $k$-mersion Theorem ([F]), the Phillips Submersion Theorem ([P]) and the general theorems due to Gromov ([G1]) and du Plessis ([duP1], [duP2] and [duP3]). Furthermore, we should refer to the homotopy principle on the $1$-jet level for fold-maps due to Èliašberg ([E1] and [E2]) (see further references in [G2]).

Let $C_{\mathcal{O}}^{\infty}(N,P)$ denote the space consisting of all $\mathcal{O}$-regular maps, $N\rightarrow P$ equipped with the $C^{\infty}$-topology. Let $\Gamma_{\mathcal{O}}(N,P)$ denote the space consisting of all continuous sections of the fiber bundle $\pi_{N}^{k}|\mathcal{O}(N,P):\mathcal{O}(N,P)\rightarrow N$ equipped with the compact-open topology. Then there exists a continuous map $j_{\mathcal{O}}:C_{\mathcal{O}}^{\infty}(N,P)\rightarrow\Gamma_{\mathcal{O}}(N,P)$ defined by $j_{\mathcal{O}}(f)=j^{k}f$. If the following property (h-P) holds, then we say in this paper that the relative homotopy principle on the existence level holds for $\mathcal{O}$-regular maps.

(h-P) Let $C$ be a closed subset of $N$ with $\partial N=\varnothing$. Let $s$ be a section in $\Gamma_{\mathcal{O}}(N,P)$ which has an $\mathcal{O}$-regular map $g$ defined on a neighborhood of $C$ to $P$, where $j^{k}g=s$. Then there exists an $\mathcal{O}$-regular map $f:N\rightarrow P$ such that $s$ and $j^{k}f$ are homotopic relative to a neighborhood of $C$ by a homotopy $s_{\lambda}$ in $\Gamma_{\mathcal{O}}(N,P)$ with $s_{0}=s$ and $s_{1}=j^{k}f$.

As important applications of [An7, Theorem 0.1] we will prove the following relative homotopy principles in (h-P). Here, $\Sigma^{n-p+1,0}(n,p)$ refers to  the space consisting of all fold jets in $J^{k}(n,p)$.

Theorem 1.1

Let $n$ and $p$ be positive integers with $n\geqq p\geqq 2$ or $n<p$. Let $k$ be a positive integer with $k\geqq n-|n-p|+2$. Let $\mathcal{O}(n,p)$ denote a $\mathcal{K}$-invariant open subspace of $J^{k}(n,p)$ containing all regular jets such that if $n\geqq p\geqq 2$, then $\mathcal{O}(n,p)$ contains $\Sigma^{n-p+1,0}(n,p)$ at least. Let $N$ and $P$ be connected smooth manifolds of dimensions $n$ and $p$ respectively with $\partial N=\emptyset$. Let $C$ be a closed subset of $N$. Let $s$ be a section in $\Gamma_{\mathcal{O}}(N,P)$ which has an $\mathcal{O}$-regular map $g$ defined on a neighborhood of $C$ to $P$, where $j^{k}g=s$.

Then there exists an $\mathcal{O}$-regular map $f:N\rightarrow P$ such that $j^{k}f$ is homotopic to $s$ relative to a neighborhood of $C$ as sections in $\Gamma_{\mathcal{O}}(N,P)$.

Let $\rho$ be an integer with $\rho\geqq 1$. Let $W_{\rho}^{k}$ denote the subset consisting of all $z\in J^{k}(n,p)$ such that the codimension of $\mathcal{K}z$ in $J^{k}(n,p)$ is not less than $\rho$ ($k$ may be $\infty$). Let $\mathcal{O}_{\ell}^{k}(n,p)$ denote a $\mathcal{K}$-invariant nonempty open subset of $J^{k}(n,p)\backslash W_{\ell+1}^{k}$. By applying Theorem 1.1 we will prove the following theorem.

Theorem 1.2

Let $\ell$ be a positive integer. Let $k\geqq\max\{\ell+1,n-|n-p|+2\}$ or $k=\infty$. Let $\mathcal{O}_{\ell}^{k}(n,p)$ denote a $\mathcal{K}$-invariant open subspace of $J^{k}(n,p)$ containing all regular jets such that if $n\geqq p\geqq 2$, then $\mathcal{O}_{\ell}^{k}(n,p)$ contains $\Sigma^{n-p+1,0}(n,p)$ at least. Then the relative homotopy principle in (h-P) holds for $\mathcal{O}_{\ell}^{k}$-regular maps.

It is well known that any smooth map $f:N\rightarrow P$ is homotopic to a smooth map $g:N\rightarrow P$ such that $j_{x}^{\infty}g$ is of finite $\mathcal{K}$-codimension for any $x\in N$ (see, for example, [W, Theorem 5.1]).

There have been described many important applications of the homotopy principles in [G2]. We only refer to the recent applications of the relative homotopy principle on the existence level to the problems in topology such as the elimination of singularities and the existence of $\mathcal{O}_{l}^{k}$-regular maps in [An1-7] and [Sa] and the relation between the stable homotopy groups of spheres and higher singularities in [An4].

Let $P$ be a closed manifold of dimension $p$. Let $\mathfrak{h}(P)$ denote the group of all homotopy classes of homotopy equivalences of $P$. Let $\mathfrak{h}_{\ell}(P)$ denote the subset of $\mathfrak{h}(P)$ which consists of all homotopy classes of maps which are homotopic to $\mathcal{O}_{l}^{\infty}$-regular homotopy equivalences. In particular, $\mathfrak{h}_{0}(P)$ is the subset of all homotopy classes of maps which are homotopic to diffeomorphisms of $P$. In this paper we will prove that the following filtration

[TABLE]

is never trivial in general.

Theorem 1.3

For a given positive integer $d$, there exists a closed oriented $p$-manifold $P$ and a sequence of positive integers $\ell_{1}$, $\ell_{2},\cdots,\ell_{d}$ with $\ell_{j}<\ell_{j+1}$ for $1\leq j<d$ such that

[TABLE]

In Section 2 we will review the results on the Boardman manifolds and the fundamental properties of $\mathcal{K}$-equivalence and $\mathcal{K}$-determinacy which are necessary in this paper. In Section 3 we will recall [An7, Theorem 0.1] and apply it in the proofs of Theorems 1.1 and 1.2. In Section 4 we will study the nonexistence problem of $\mathcal{O}_{l}^{k}$-regular maps. In Section 5 we will study the filtration in (1.1) and prove Theorem 1.3.

2 Boardman manifolds and $\mathcal{K}$-orbits

Throughout the paper all manifolds are Hausdorff, paracompact and smooth of class $C^{\infty}$. Maps are basically smooth (of class $C^{\infty}$) unless otherwise stated.

For a Boardman symbol (simply symbol) $I=(i_{1},\cdots,i_{k})$ with $n\geqq i_{1}\geqq\cdots\geqq i_{k}\geqq 0$, let $\Sigma^{I}(n,p)$ denote the Boardman manifold of symbol $I$ in $J^{k}(n,p)$ which has been defined in [T], [L], [Bo] and [MaTB]. Let $A_{n}=\mathbb{R}[[x_{1},\cdots,x_{n}]]$ denote the formal power series of algebra on variables $x_{1},\cdots,x_{n}$. Let $\mathfrak{m}_{n}$ be its maximal ideal and $A_{n}(k)=A_{n}/\mathfrak{m}_{n}^{k+1}$. Let $z=j_{0}^{k}f\in J^{k}(n,p)$ where $f=(f^{1},\cdots,f^{p}):(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)$. We define $\mathcal{I}(z)$ to be the ideal in $A_{n}(k)$ generated by the image in $A_{n}(k)$ of the Taylor expansions of $f^{1},\cdots,f^{p}$. It has been proved in [Bo] and [MaTB] that the Boardman symbol $I(z)$ of $z$ depends only on the ideal $\mathcal{I}(z)$ by the notion of the Jacobian extension. Let $\Sigma^{I}(N,P)$ denote the subbundle of $J^{k}(N,P)$ over $N\times P$ associated to $\Sigma^{I}(n,p)$. Let $\Sigma_{x,y}^{I}(N,P)$ denote the fiber of $\Sigma^{I}(N,P)$ over $(x,y)\in N\times P$.

Since codim$\Sigma^{i_{1}}(n,p)=(p-n+i_{1})i_{1}$, the following proposition follows from [An6, Remark 2.1], which has been proved by using the results in [Bo, Section 6].

Proposition 2.1

Let $I=(i_{1},\cdots,i_{\ell})$ be a symbol such that $i_{1}\geqq\max\{n-p+1,1\}$ and $\Sigma^{I}(n,p)$ is nonempty. Then we have

[TABLE]

In particular, if $i_{\ell}>0$, then we have $\mathrm{codim}\Sigma^{I}(n,p)\geqq|n-p|+\ell$.

Let $\Omega^{I}(n,p)$ denote the union of all Boardman manifolds $\Sigma^{J}(N,P)$ with $J\leq I$ in the lexicographic order. We have the following lemma (see [duP1]).

Lemma 2.2

The space $\Omega^{I}(n,p)$ is open in $J^{k}(n,p)$.

Let us review the $\mathcal{K}$-equivalence of two smooth map germs $f,g:(N,x)\rightarrow(P,y)$, which has been introduced in [MaIII, (2.6)], by following [Mart, II, 1]. We say that the above two map germs $f$ and $g$ are $\mathcal{K}$-equivalent if there exists a smooth map germ $\phi:(N,x)\rightarrow GL(\mathbb{R}^{p})$ and a local diffeomorphism $h:(N,x)\rightarrow(N,x)$ such that $f(x)=\phi(x)g(h(x))$. It is known that this $\mathcal{K}$-equivalence is nothing but the contact equivalence introduced in [MaIII]. The contact group $\mathcal{K}$ is defined as a certain subgroup of the group of germs of local diffeomorphisms $(N,x)\times(P,y)$ and acts on $J_{x,y}^{k}(N,P)$. For a $k$-jet $z$ in $J_{x,y}^{k}(N,P)$ let $\mathcal{K}z$ denote the orbit of $\mathcal{K}$ through $z$. As is well known, $\mathcal{K}z$ is an orbit of a Lie group. Hence, $\mathcal{K}z$ is a submanifold of $J_{x,y}^{k}(N,P)$. This fact is also observed from the above definition. The following lemma is important in this paper.

Lemma 2.3

The Boardman manifold $\Sigma_{x,y}^{I}(N,P)$ in $J_{x,y}^{k}(N,P)$ is invariant with respect to the action of $\mathcal{K}$.

Proof. Let $z=j_{x}^{k}f$ and $w=j_{x}^{k}g$ be $k$-jets in $J_{x,y}^{k}(N,P)$ such that two map germs $f$ and $g$ are $\mathcal{K}$-equivalent as above. Let $h_{\ast}:C_{x}\rightarrow C_{x}$ be the isomorphism defined by $h_{\ast}(\phi)=\phi\circ h$. By the definition of $\mathcal{K}$-equivalence we have $h_{\ast}(\mathfrak{I}(g))=\mathfrak{I}(f)$. The Thom-Boardman symbols of $j_{x}^{k}f$ and $j_{x}^{k}g$ are determined by $\mathfrak{I}(f)$ and $\mathfrak{I}(g)$, and are the same by [MaTB, 2, Corollary]. This proves the assertion.

Let us review the results in [MaIII], [MaIV] and [MaV] which are necessary in this paper. Let $C^{\infty}(N,x)$ and $C^{\infty}(P,y)$ denote the rings of smooth function germs on $(N,x)$ and $(P,y)$ respectively. Let $\mathfrak{m}_{x}$ and $\mathfrak{m}_{y}$ denote their maximal ideals respectively. Let $f:(N,x)\rightarrow(P,y)$ be a germ of a smooth map. Let $f^{\ast}:C^{\infty}(P,y)\rightarrow C^{\infty}(N,x)$ denote the homomorphism defined by $f^{\ast}(a)=a\circ f$. Let $\theta(N)_{x}$ denote the $C^{\infty}(N,x)$-module of all germs at $x$ of smooth vector fields on $(N,x).$ We define $\theta(P)_{y}$ similarly for $y\in P$. Let $\theta(f)_{x}$ denote the $C^{\infty}(N,x)$-module of germs at $x$ of smooth vector fields along $f$, namely which consists of all smooth germs $\varsigma:(N,x)\rightarrow TP$ such that $p_{P}\circ\varsigma=f$. Here, $p_{P}:TP\rightarrow P$ is the canonical projection. Then we have the homomorphisms

[TABLE]

defined by $tf(u_{N})=df\circ u_{N}$ for $u_{N}\in\theta(N)_{x}$. For a singular jet $z=j_{0}^{k}f\in J^{k}(N,P)$ there has been defined the isomorphism

[TABLE]

in [MaIII, (7.3)] such that $T_{z}(\mathcal{K}z)$ corresponds to $tf(\mathfrak{m}_{x}\theta(N)_{x})+f^{\ast}(\mathfrak{m}_{y})(\theta(f)_{x})$ modulo $\mathfrak{m}_{x}^{k+1}\theta(f)_{x}$. We do not here explain the definition. According to [MaIII] we define $d(f,\mathcal{K})$ to be

[TABLE]

which is equal to codim$\mathcal{K}z$.

3 Proofs of Theorems 1.1 and 1.2.

In this section we prove Theorems 1.1 and 1.2.

Let $k$ be a positive integer. Let $W_{\rho}^{k}=W_{\rho}^{k}(n,p)$ denote the subset consisting of all $z\in J^{k}(n,p)$ such that the codimension of $\mathcal{K}z$ in $J^{k}(n,p)$ is not less than $\rho$. The following lemma has been observed in [MaV, Section 7 and Proof of Theorem 8.1].

Lemma 3.1

Let $\rho$ be an integer with $\rho\geqq 1$. Then $W_{\rho}^{k}$ is an algebraic subset of $J^{k}(n,p)$.

The order of $\mathcal{K}$-determinacy is estimated by the codimension of a $\mathcal{K}$-orbit as follows.

Proposition 3.2

Let $k$ be an integer with $k>\rho$. Let $z=j^{k}f$ be a singular jet in $J^{k}(n,p)\backslash W_{\rho+1}^{k}$. Then $z$ is $\mathcal{K}$-$k$-determined.

Proof. It follows from [W, Theorem 1.2 (iii)] that if $d=$codim$\mathcal{K}z$, then $z$ is $\mathcal{K}$-$(d+1)$-determined. Hence, if $z\in J^{k}(n,p)\backslash W_{\rho+1}^{k}$, then $d\leq\rho$ and $z$ is $\mathcal{K}$-$k$-determined.

We define the bundle homomorphism

[TABLE]

Let $w=j_{x}^{k}f\in J_{x,y}^{k}(N,P)$ and $z=\pi_{k}^{k}(w)$. Then we have $j^{k-1}f:(N,x)\rightarrow(J^{k-1}(N,P),z)$ and $d(j^{k-1}f):T_{x}N\rightarrow T_{z}(J^{k-1}(N,P))$. We set

[TABLE]

Let $I^{\prime}$ be a symbol of length $k$. Let $\mathbf{K}(\Sigma^{I^{\prime}})$ denote the kernel subbundle of $(\pi_{N}^{k}|\Sigma^{I^{\prime}}(N,P))^{\ast}(TN)$ defined by

[TABLE]

The following theorem follows from the corresponding assertion for the case $k=\infty$ in [B, (7.7)]. This is very important in the proof of Theorem 1.1.

Theorem 3.3

If $I^{\prime}=(i_{1},\cdots,i_{k-2},0,0)$ and $I=(i_{1},\cdots,i_{k-2,}0)$, then we have

[TABLE]

for any $w\in\Sigma^{I^{\prime}}(N,P)$.

Let us review a general condition on $\mathcal{O}(n,p)$ for the relative homotopy principle on the existence level in [An7]. We say that a nonempty $\mathcal{K}$-invariant open subset $\mathcal{O}(n,p)$ is admissible if $\mathcal{O}(n,p)$ consists of all regular jets and a finite number of disjoint $\mathcal{K}$-invariant nonempty submanifolds $V^{i}(n,p)$ of codimension $\rho_{i}$ ($1\leq i\leq\iota$) such that the following properties (H-i) to (H-v) are satisfied.

(H-i) $V^{i}(n,p)$ consists of singular $k$-jets of rank $r_{i}$, namely, $V^{i}(n,p)\subset\Sigma^{n-r_{i}}(n,p)$.

(H-ii) For each $i$, the set $\mathcal{O}(n,p)\backslash\{\cup_{j=i}^{\iota}V^{j}(n,p)\}$ is an open subset.

(H-iii) For each $i$ with $\rho_{i}\leq n$, there exists a $\mathcal{K}$-invariant submanifold $V^{i}(n,p)^{(k-1)}$ of $J^{k-1}(n,p)$ such that $V^{i}(n,p)$ is open in $(\pi_{k-1}^{k})^{-1}(V^{i}(n,p)^{(k-1)})$.

(H-iv) If $n\geqq p\geqq 2$, then $V^{1}(n,p)=\Sigma^{n-p+1,0}(n,p)$.

Here, $\Sigma^{n-p+1,0}(n,p)$ denotes the Thom-Boardman manifold in $J^{k}(n,p)$, which consists of $\mathcal{K}$-orbits of fold jets. Let $V^{i}(N,P)$ denote the subbundle of $J^{k}(N,P)$ associated to $V^{i}(n,p)$. Let $\mathbf{K}(V^{i})$ be the kernel bundle in $(\pi_{N}^{k})^{\ast}(TN)|_{V^{i}(N,P)}$ defined by $\mathbf{K}(V^{i})_{z}=(z,\mathrm{Ker}(d_{x}f))$.

(H-v) For each $i$ with $\rho_{i}\leq n$ and any $z\in V^{i}(N,P)$, we have

[TABLE]

Then we have proved the following theorem in [An7, Theorem 0.1].

Theorem 3.4

Let $k\geqq n-|n-p|+2$. Let $n\geqq p\geqq 2$ or $n<p$. Let $\mathcal{O}(n,p)$ denote an admissible open subspace of $J^{k}(n,p)$. Then the relative homotopy principle in (h-P) holds for $\mathcal{O}$-regular maps.

We set

[TABLE]

Let $J=(j_{1},\cdots,j_{k})$ be a symbol of a singular jet with codim$\Sigma^{J}(n,p)\leq n$. If $k\geqq n-|n-p|+2$, we have by Proposition 2.1 that $i_{k-1}=i_{k}=0$. Indeed, if $i_{k-1}>0$, then

[TABLE]

So we set $J=(j_{1},\cdots,j_{k-2},0,0)$, $J^{\ast}=(i_{1},\cdots,i_{k-2,}0)$ and

[TABLE]

Lemma 3.5

Let $J=(j_{1},\cdots,j_{k-2},0,0)$ and $J^{\ast}=(j_{1},\cdots,j_{k-2,}0)$ be as above. Then $V_{J}(n,p)$ is open in $(\pi_{k-1}^{k})^{-1}(V_{J^{\ast}}(n,p)^{(k-1)})$.

Proof. It is evident that

[TABLE]

So we have $V_{J}(n,p)\subset(\pi_{k-1}^{k})^{-1}(V_{J^{\ast}}(n,p)^{(k-1)})$. Since $\pi_{k-1}^{k}$ is an open map, we have that $V_{J}(n,p)$ is an open subset of $(\pi_{k-1}^{k})^{-1}(V_{J^{\ast}}(n,p)^{(k-1)})$.

Let us prove Theorem 1.1.

Proof of Theorem 1.1. By Theorem 3.4 it is enough to prove that $\mathcal{O}(n,p)$ is admissible. Let $J$ be a symbol of length $k$. By Lemma 2.3, $V_{J}(n,p)$ is $\mathcal{K}$-invariant. We have that

(H1) $\mathcal{O}(n,p)$ is decomposed into a finite union of all $V_{J}(n,p)$,

(H2) For each symbol $J$, the set $\mathcal{O}(n,p)\cap\Omega^{J}(n,p)$ is an open subset of $\mathcal{O}(n,p)$,

(H3) $V_{J}(n,p)$ is open in $(\pi_{k-1}^{k})^{-1}(V_{J^{\ast}}(n,p)^{(k-1)})$ by lemma 3.5,

(H4) If $n\geqq p\geqq 2$, then $\mathcal{O}(n,p)\supset\Sigma^{n-p+1,0}(n,p)$ by the assumption,

(H5) Property (3.2) holds for $V_{J}(n,p)$ by Theorem 3.3 and Lemma 3.5.

Since $\mathcal{O}(n,p)$ satisfies the properties (H1) to (H5), we have proved Theorem 1.1.

We next prove Theorem 1.2.

Proof of Theorem 1.2. If $\ell$ is finite, then it follows from Lemma 3.2 that if $k>\ell$, then any $k$-jet $z$ of $J^{k}(n,p)\backslash W_{\ell+1}^{k}$ is $\mathcal{K}$-$k$-determined and we have

[TABLE]

Therefore, if $k\geqq\max\{\ell+1,n-|n-p|+2\}$, then the relative homotopy principle in (h-P) holds for $\mathcal{O}_{\ell}^{k}$-regular maps by Theorem 1.1 and also for $\mathcal{O}_{\ell}^{\infty}$-regular maps.

Corollary 3.6

Under the same assumption of Theorem 1.2, given a map $f:N\rightarrow P$ is homotopic to an $\mathcal{O}_{\ell}^{k}$-regular map if and only if there exists a section $s\in\Gamma_{\mathcal{O}_{\ell}^{k}}(N,P)$ such that $\pi_{P}^{k}\circ s$ is homotopic to $f$.

Corollary 3.7

Let $\mathfrak{h}_{\ell}(P)$ be as in Introduction. Then the homotopy class of a homotopy equivalence $f:P\rightarrow P$ lies in $\mathfrak{h}_{\ell}(P)$ if and only if $j^{\infty}f$ is homotopic to a section in $\Gamma_{\mathcal{O}_{\ell}^{\infty}}(N,P)$.

Here we give two remarks.

Remark 3.8

Let $W_{\infty}^{\infty}$ denote the subspace of $J^{\infty}(n,p)$ which consists of all jets $z$ such that any smooth map germ $f$ with $z=j^{\infty}f$ is not finitely determined. Let $W_{\infty}^{\infty}(N,P)$ is the subbundle of $J^{\infty}(N,P)$ associated to $W_{\infty}^{\infty}$. It has been proved (see, for example, [W, Theorem 5.1]) that $W_{\infty}^{\infty}$ is not of finite codimension in $J^{\infty}(n,p)$. Consequently, the space of all smooth maps $f:N\rightarrow P$ with $j^{\infty}f(N)\subset J^{\infty}(N,P)\backslash W_{\infty}^{\infty}(N,P)$ is dense in $C^{\infty}(N,P)$. In other words if $N$ is compact, then a smooth map $f:N\rightarrow P$ has an integer $\ell$ such that $f$ is homotopic to an $O_{\ell}^{\infty}$-regular map.

Remark 3.9

It is very important to study the topology of the space $W_{\ell+1}^{k}(n,p)$ and obstructions for finding an $O_{\ell}^{k}$-regular map. The Thom polynomials related to $W_{\ell+1}^{k}(n,p)$ have been studied in the dimensions $n=p\leqq 8$ in [O] and [F-R].

4 Nonexistence theorems

In this section we will discuss the nonexistence of $\mathcal{O}_{\ell}^{k}$-regular maps $f:N\rightarrow P$. Let $W_{\ell+1}^{k}(N,P)$ denote the subbundle of $J^{k}(N,P)$ associated to $W_{\ell+1}^{k}(n,p)$. By the homotopy principle for $\mathcal{O}_{\ell}^{k}$-regular maps in Theorem 1.2, the existence of a section of $J^{k}(N,P)\backslash W_{\ell+1}^{k}(N,P)$ over $N$ is equivalent to the existence of an $\mathcal{O}_{\ell}^{k}$-regular map. However, it is not so easy to find obstructions associated to $W_{\ell+1}^{k}(N,P)$ such as Thom polynomials of $W_{\ell+1}^{k}(N,P)$, and so we will adopt a method applied in [An1], [I-K] and [duP4] in this section.

For $k\geqq p+1$, let $\Sigma(n,p;k)$ denote the algebraic subset of all $C^{\infty}$-nonstable $k$-jets of $J^{k}(n,p)$ defined in [MaV]. Note that for $k^{\prime}>k$, $(\pi_{k}^{k^{\prime}})^{-1}(\Sigma(n,p;k))=\Sigma(n,p;k^{\prime})$. We have proved the following proposition in [An1, Corollary 5.6].

Proposition 4.1

Let $k\geqq p+1$. If

[TABLE]

then we have that $\Sigma^{i}(n,p)\subset\Sigma(n,p;k)$.

In [I-K] the following proposition has been proved, while it has not been stated explicitly and the proof has been given in the context without the details. So we give a sketchy proof.

Proposition 4.2** ([I-K])**

Let $\ell$ be a nonnegative integer and $k\geqq p+\ell+1$. If

[TABLE]

then we have that $\Sigma^{i}(n,p)\subset W_{\ell+1}^{k}(n,p)$. In particular, if $n=p$ and $\frac{1}{2}i^{2}(i-1)\geqq n+\ell,$ then we have that $\Sigma^{i}(n,n)\subset W_{\ell+1}^{k}(n,n).$

Proof. Take a jet $z$ in $\Sigma^{i}(n,p)$ such that $z=j_{0}^{k}f$. Suppose that $z\notin W_{\ell+1}^{k}$, and hence codim$\mathcal{K}z\leqq\ell$. By [MaIV] there exists a versal unfolding $F:(\mathbb{R}^{n}\times\mathbb{R}^{\ell},0)\rightarrow(\mathbb{R}^{p}\times\mathbb{R}^{\ell},0)$ of $f$ and $j_{(0,0)}^{k}F\notin\Sigma(n+\ell,p+\ell;k)$. Here, we note that $j_{(0,0)}^{k}F$ is of kernel rank $i$. By the assumption and Proposition 4.1 we have

[TABLE]

This implies $j_{(0,0)}^{k}F\in\Sigma(n+\ell,p+\ell;k)$. This is a contradiction. Hence, $z$ lies in $W_{\ell+1}^{k}$.

We show the following proposition by applying Proposition 4.2.

Proposition 4.3

Let $\ell$ be a nonnegative integer and $k\geqq p+\ell+1$. If $\Sigma^{i}(n,p)\subset W_{\ell+1}^{k}(n,p)$, then we have that for any positive integer $m$, $\Sigma^{i}(m+n,m+p)\subset W_{\ell+1}^{k}(m+n,m+p)$.

Proof. Let $z=j_{0}^{k}f\in\Sigma^{i}(m+n,m+p)$. Setting $\alpha=j_{0}^{1}f$, we identify $\alpha$ with the homomorphism $\mathbb{R}^{m+n}\rightarrow\mathbb{R}^{m+p}$. Let $\mathrm{Ker}(\alpha)^{\perp}$ and $\mathrm{\operatorname{Im}}(\alpha)^{\perp}$ be the orthogonal complement of the kernel $\mathrm{Ker}(\alpha)$ and the image $\mathrm{\operatorname{Im}}(\alpha)$ of $\alpha$ respectively. Let $L$ and $M$ be subspaces of $\mathrm{Ker}(\alpha)^{\perp}$ and $\mathrm{\operatorname{Im}}(\alpha)$ of dimension $m$ such that $\alpha$ maps $L$ onto $M$ isomorphically. Let $L^{\perp}$ and $M^{\perp}$ be their orthogonal complements in $\mathrm{Ker}(\alpha)^{\perp}$ and $\mathrm{\operatorname{Im}}(\alpha)$ respectively. Then $\alpha$ is decomposed as in the following exact sequence.

[TABLE]

Let us choose coordinates

[TABLE]

of $L$, $L^{\perp}$ and $\mathrm{Ker}(\alpha)$, and coordinates

[TABLE]

of $M$, $M^{\perp}$ and $\mathrm{\operatorname{Im}}(\alpha)^{\perp}$ respectively. Since $\alpha$ maps $L$ onto $M$ isomorphically, there exist the new coordinates $(x_{1},\cdots,x_{m+n})$ of $\mathbb{R}^{m+n}$ such that

[TABLE]

and that

[TABLE]

Setting $\overset{\bullet}{x}=(x_{m+1},\cdots,x_{m+n})$, we define the map $g:(\mathbb{R}^{n},0)\rightarrow(\mathbb{R}^{p},0)$ by

[TABLE]

Then $f$ is an unfolding of $g$ by (4.1) and $g$ is of kernel rank $i$ at the origin. We next prove by following the argument and the notation used in [MaIV, Section 1] that $d(g,\mathcal{K})$ is equal to $d(f,\mathcal{K})$. Define $\pi:\theta(f)\rightarrow\theta(g)$ by

[TABLE]

where $a_{j}\in C^{\infty}(\mathbb{R}^{m+n},0)$, $a_{j}^{\prime}\in C^{\infty}(\mathbb{R}^{n},0)$ and $a_{j}^{\prime}(\overset{\bullet}{x})=a_{j}(0,\cdots,0,\overset{\bullet}{x})$. We note that

[TABLE]

Since

[TABLE]

for some $b_{j}\in C^{\infty}(\mathbb{R}^{m+n},0)$, we have

[TABLE]

Hence, the assertion follows from an elementary calculation under the definition in (2.3).

Since $j_{0}^{k}g\in\Sigma^{i}(n,p)\subset W_{\ell+1}^{k}(n,p)$, we have $d(g,\mathcal{K})\geqq\ell+1$. Hence, we have $d(f,\mathcal{K})\geqq\ell+1$. This shows $z\in W_{\ell+1}^{k}(m+n,m+p)$. This is what we want.

Let $\xi$ be a stable vector bundle over a space. Let $\mathbf{c}(\Sigma^{i},\xi)$ denote the determinant of the $(p-n+i)$-matrix whose $(s,t)$-component is the $(i+s-t)$-th Stiefel-Whitney class $W_{i+s-t}(\xi)$. If $n-p$ and $i$ are even, say $n-p=2u$ and $i=2v$, and if $\xi$ is orientable, then $\mathbf{c}_{\mathbb{Z}}(\Sigma^{i},\xi)$ expresses the determinant of the $(v-u)$-matrix whose $(s,t)$-component is the $(v+s-t)$-th Pontrjagin class $P_{v+s-t}(\xi)$.

[TABLE]

Let $\tau_{X}$ denote the stable tangent bundle of a manifold $X$. If $f:N\rightarrow P$ is a smooth map transverse to $\Sigma^{i}(N,P)$ and $\xi=\tau_{N}-f^{\ast}(\tau_{P})$, then $\mathbf{c}(\Sigma^{i},\xi)$ (resp. $\mathbf{c}_{\mathbb{Z}}(\Sigma^{i},\xi)$) is equal to the (resp. integer) Thom polynomial of the topological closure of $(j^{k}f)^{-1}(\Sigma^{i}(N,P))$ ([Po], [Ro] and see also [An1, Proposition 5.4]). If it does not vanish, then $(j^{k}f)^{-1}(\Sigma^{i}(N,P))$ cannot be empty by the obstruction theory in [St]. Hence, we have the following corollary of Propositions 4.2 and 4.3.

Corollary 4.4

Let $f:M\rightarrow Q$ be a smooth map with $\dim M=m+n$ and $\dim Q=m+p$. Under the same assumption of Proposition 4.2. we assume that either

(i) $\mathbf{c}(\Sigma^{i},\tau_{M}-f^{\ast}(\tau_{Q}))$ does not vanish, or

(ii) $M$ and $\tau_{M}-f^{\ast}(\tau_{Q})$ are orientable, $n-p$ and $i$ are even and$\ \mathbf{c}_{\mathbb{Z}}(\Sigma^{i},\tau_{M}-f^{\ast}(\tau_{Q}))$ does not vanish.

Then $f$ is not homotopic to any $\mathcal{O}_{\ell}^{k}$-regular map.

5 Homotopy equivalences

In this section we will study the filtration in (1.1) in Introduction by applying Corollaries 3.7 and 4.4 and Remark 3.8.

Let us first review what is called the Sullivan’s exact sequence in the surgery theory following [M-M] (see also [K-M], [Su] and [Br]).

In what follows $P$ is a closed and oriented $n$-manifold. We define the set $\mathcal{S}(P)$ to be the set of all equivalence classes of homotopy equivalences $f:N\rightarrow P$ of degree $1$ under the following equivalence relation. Let $N_{j}$ be closed oriented $n$-manifolds and let $f_{j}:N_{j}\rightarrow P$ be homotopy equivalences of degree $1$ ($j=1,2$). We say that $f_{1}$ and $f_{2}$ are equivalent if there exists an $h$-cobordism $W$ of $N_{1}$ and $N_{2}$ and a homotopy equivalence $F:(W,N_{1}\cup(-N_{2}))\rightarrow(P\times[0,1],P\times 0\cup(-P)\times 1)$ of degree $1$ such that $F|N_{j}=f_{j}$ ($j=1,2$).

Let $O(k)$ denote the rotation group of $\mathbb{R}^{k}$ and let $G_{k}$ denote the space of all homotopy equivalence of the $(k-1)$-sphere $S^{k-1}$ equipped with the compact-open topology. By considering the canonical inclusions $O(k)\rightarrow O(k+1)$ and $G_{k}\rightarrow G_{k+1}$, we set $O=\lim_{k\rightarrow\infty}O(k)$ and $G=\lim_{k\rightarrow\infty}G_{k}$. Let $BO$ and $BG$ denote the classifying spaces for $O$ and $G$. Then we have the canonical maps $\pi(m):BO(m)\rightarrow BG(m)$ and $\pi:BO\rightarrow BG$, which are regarded as fibrations with fibers $G(m)/O(m)$ and $G/O$ respectively. For a sufficiently large number $m$, let $\eta_{O(m)}$ denote the universal vector bundle over $BO(m)$ and let $i_{G/O}:G(m)/O(m)\rightarrow BO(m)$ be the inclusion of a fiber. Set $\eta_{G/O}=(i_{G/O})^{\ast}\eta_{O(m)}$. Then $\eta_{G/O}$ has a trivialization $t_{G/O}:\eta_{G/O}\rightarrow\mathbb{R}^{m}$ as a spherical fibration.

We next recall the surgery obstruction $\mathfrak{s}_{4q}^{P}:[P,G/O]\rightarrow\mathbb{Z}$ only in the case of $n=4q$. For $[\alpha]\in[P,G/O]$ let $\eta=\alpha^{\ast}(\eta_{G/O})$ with the canonical bundle map $\overline{\alpha}:\eta\rightarrow\eta_{G/O}$ covering $\alpha$ and the projection $\pi_{\eta}$ onto $P$. We deform $t_{G/O}\circ\overline{\alpha}$ to a map transverse to $0\in\mathbb{R}^{m}$ and let $M$ be the inverse image of [math] with a map $\pi_{\eta}|M:M\rightarrow P$ of degree $1$. We define $\mathfrak{s}_{4q}^{P}([\alpha])=(1/8)(\sigma(M)-\sigma(P))$. If $P$ is simply connected in addition, then there have been defined an injection $j^{P}:\mathcal{S}(P)\rightarrow[P,G/O]$ such that if $\mathfrak{s}_{4q}^{P}([\alpha])=0$, $\pi_{\eta}|M$ is deformed to a homotopy equivalence $f:N\rightarrow P$ of degree $1$ under a certain cobordism. The following is the Sullivan’s exact sequence.

[TABLE]

Let us recall the cobordism group $\Omega_{n}^{h-eq}$ of homotopy equivalences of degree $1$ in [An5]. Let $N_{j}$ and $P_{j}$ be oriented closed $n$-manifolds and let $f_{j}:N_{j}\rightarrow P_{j}$ be homotopy equivalences of degree $1$ ($j=1,2$). We say that $f_{1}$ and $f_{2}$ are cobordant if there exists an oriented $(n+1)$-manifold $W$, $V$ and a homotopy equivalence $F:(W,\partial W)\rightarrow(V,\partial V)$ of degree $1$ such that $\partial W=N_{1}\cup(-N_{2})$, $\partial V=P_{1}\cup(-P_{2})$ and $F|N_{j}=f_{j}$. The cobordism class of $f:N\rightarrow P$ is denoted by $[f:N\rightarrow P]$. Let $\Omega_{n}^{h-eq}$ denote the set which consists of all cobordism classes of homotopy equivalences of degree $1$. We provide $\Omega_{n}^{h-eq}$ with a module structure by setting

$\bullet$ $[f_{1}:N_{1}\rightarrow P_{1}]+[f_{2}:N_{2}\rightarrow P_{2}]=[f_{1}\cup f_{2}:N_{1}\cup N_{2}\rightarrow P_{1}\cup P_{2}]$,

$\bullet$ $-[f:N\rightarrow P]=[f:(-N)\rightarrow(-P)].$

The null element is defined to be $[f:N\rightarrow P]$ which bound a homotopy equivalence $F:(W,\partial W)\rightarrow(V,\partial V)$ of degree $1$ such that $\partial W=N$, $\partial V=P$ and $F|N=f$. Even if $P$ is not simply connected, we can find $f_{1}:N_{1}\rightarrow P_{1}$ with $P_{1}$ being simply connected in the same cobordism class by killing $\pi_{1}(N)\approx\pi_{1}(P)$ by usual surgery.

Let $\mathbf{c}_{\mathbb{Q}}(\Sigma^{2i},\eta_{G/O})$ denote the image of $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i},\eta_{G/O})$ in $H^{4i^{2}}(G/O;\mathbb{Q})$. Let $\alpha=j^{P}([f:N\rightarrow P])$ and let $c_{P}:P\rightarrow BSO$ be a classifying map of the tangent bundle $TP$ of $P$. Then it induces the homomorphism $\mathcal{C}_{2i}:{\Omega}_{4q}^{h-eq}\rightarrow H_{4q-4i^{2}}(G/O;\mathbb{Q})$ defined by

[TABLE]

under the identification

[TABLE]

in $\Sigma_{j=0}^{q-i^{2}}H_{4j}(G/O;\mathbb{Q})\otimes H_{4q-4i^{2}-4j}(BSO;\mathbb{Q})$. We have that

[TABLE]

Furthermore, we have proved in [An5, Theorems 3.2 and 4.1] that for integers $q$ and $i$ with $q\geqq i^{2}\geqq 1$,

[TABLE]

The following theorem follows from (5.1), Proposition 4.2 and Corollary 4.4.

Theorem 5.1

Let $\ell$, $q$ and $i$ be integers with $\ell\geqq 0$ and $q\geqq i^{2}$. Let $k\geqq 4q+\ell+1$. There exists a cobordism class $[f:N\rightarrow P]\in\Omega_{4q}^{h-eq}$ such that $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i},\tau_{N}-f^{\ast}(\tau_{P}))$ is not a torsion element and that if $4i^{3}-2i^{2}\geqq 4q+\ell\geqq 4i^{2}+\ell$, then $f$ is not cobordant in $\Omega_{4q}^{h-eq}$ to any $\mathcal{O}_{\ell}^{k}$-regular map.

We can prove the following theorem using Theorem 5.1 by applying the same argument in the proof of [An5, Theorem 0.2]. However, Theorem 1.2 is very important in the following and the situation is rather different. Therefore, we give its proof.

Theorem 5.2

Let $\ell$, $q$ and $i$ be given integers with $\ell\geqq 0$ and $q\geqq i^{2}$. Let $k\geqq 8q+\ell+1$. If $4i^{3}-2i^{2}\geqq 4q+\ell\geqq 4i^{2}+\ell$, then there exists a closed connected oriented $8q$-manifold $P$ and a homotopy equivalence $f:P\rightarrow P$ of degree $1$ such that $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i},\tau_{P}-f^{\ast}(\tau_{P}))\neq 0$ and that $f$ is not cobordant in $\Omega_{8q}^{h-eq}$ to any $\mathcal{O}_{\ell}^{k}$-regular homotopy equivalence of degree $1$.

Proof. It follows from Theorem 5.1 that there exists a homotopy equivalence $f:N\rightarrow P$ of degree $1$ between $4q$-manifolds such that $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i},\tau_{N}-f^{\ast}(\tau_{P}))$ is not a torsion element. Let $f^{-1}:P\rightarrow N$ be a homotopy inverse of $f$. Define $g:N\times P\rightarrow N\times P$ by $g(x,y)=(f^{-1}(y),f(x))$. We have $k\geq\dim N\times P+\ell+1$. If we prove that $\mathbf{c_{\mathbb{Z}}}(\Sigma^{2i},\tau_{N\times P}-g^{\ast}(\tau_{N\times P}))$ does not vanish, then, by Corollary 4.4, $g$ is not homotopic to any $\mathcal{O}_{\ell}^{k}$-regular map. We set $\xi=\tau_{N\times P}-g^{\ast}(\tau_{N\times P})=\tau_{N}\times\tau_{P}-f^{\ast}(\tau_{P})\times(f^{-1})^{\ast}(\tau_{N})$. Then

[TABLE]

modulo torsion in $H^{\ast}(N;\mathbb{Z})\otimes H^{\ast}(P;\mathbb{Z})$. The term of $p_{j}(\xi)$ which lies in $H^{4j}(N;\mathbb{Z})\otimes H^{0}(P;\mathbb{Z})$ is equal modulo torsion to

[TABLE]

Hence, we have that $\mathbf{c_{\mathbb{Z}}}(\Sigma^{2i},\tau_{N\times P}-g^{\ast}(\tau_{N\times P})$ is equal to the sum of $\mathbf{c_{\mathbb{Z}}}(\Sigma^{2i},\tau_{N}-f^{\ast}(\tau_{P}))\otimes 1$ and the other term which lies in $\Sigma_{j=1}^{i^{2}}H^{4i^{2}-4j}(N;\mathbb{Z})\otimes H^{4j}(P;\mathbb{Z})$ modulo torsion. Since $\mathbf{c_{\mathbb{Z}}}(\Sigma^{2i},\tau_{N}-f^{\ast}(\tau_{P}))$ does not vanish, it follows that $\mathbf{c_{\mathbb{Z}}}(\Sigma^{2i},\tau_{N\times P}-g^{\ast}(\tau_{N\times P}))$ does not vanish. This completes the proof.

We are now ready to prove Theorem 1.3.

Proof of Theorem 1.3. In the proof $k$ refers to a sufficiently large integer. Let $i_{0}=2$, which is the smallest integer such that $4i^{3}-2i^{2}\geqq 4i^{2}$ with $q=4$ and $\ell=8$. Then we have, by Theorem 5.2, a closed connected oriented $8\cdot 4$-manifold $P_{0}$ and a homotopy equivalence $f_{0}:P_{0}\rightarrow P_{0}$ of degree $1$ such that $\mathbf{c}_{\mathbb{Z}}(\Sigma^{4},\tau_{P_{0}}-f_{0}^{\ast}(\tau_{P_{0}}))\neq 0$ and that $f_{0}$ is not homotopic to any $\mathcal{O}_{8}^{k}$-regular map. By Remark 3.8 there exists an integer $\ell$ such that $f_{0}$ is homotopic to an $\mathcal{O}_{\ell}^{k}$-regular map. Let $\ell_{1}$ be such a smallest integer.

We assume the following (A-$t$) for an integer $t\geqq 0$, where $\ell_{0}=8$.

(A-$t$) We have constructed integers $\ell_{t}$, $\ell_{t+1}$, $i_{t}$, a closed oriented $8\cdot i_{t}^{2}$-manifold $P_{t}$ and an $\mathcal{O}_{\ell_{t+1}}^{k}$-regular homotopy equivalence $f_{t}:P_{t}\rightarrow P_{t}$ of degree $1$ such that $4i_{t}^{3}-2i_{t}^{2}\geqq 4i_{t}^{2}+\ell_{t}$, $\ell_{t+1}>\ell_{t}$, $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i_{t}},\tau_{P_{t}}-f_{t}^{\ast}(\tau_{P_{t}}))\neq 0$ and that $f_{t}$ is not homotopic to any $\mathcal{O}_{\ell_{t}}^{k}$-regular map.

Under the assumption (A-$t$) we prove (A-$(t+1)$) with $\ell_{t+1}<\ell_{t+2}$. Let $i_{t+1}$ be the smallest integer among the integers $i>0$ with $4i^{3}-2i^{2}\geqq 4i^{2}+\ell_{t+1}$. Then it follows from Theorem 5.2 that there exist a closed connected oriented $8\cdot i_{t+1}^{2}$-manifold $P_{t+1}$ and a homotopy equivalence $f_{t+1}:P_{t+1}\rightarrow P_{t+1}$ of degree $1$ such that $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i_{t}},\tau_{P_{t+1}}-f_{t+1}^{\ast}(\tau_{P_{t+1}}))\neq 0$ and that $f_{t+1}$ is not homotopic to any $\mathcal{O}_{\ell_{t+1}}^{k}$-regular map. It follows Remark 3.8 that there exists an integer $\ell$ such that $f_{t+1}$ is homotopic to an $\mathcal{O}_{\ell}^{k}$-regular map. Let $\ell_{t+2}$ be the smallest integer among those integers $\ell$. Hence, we have $\ell_{t+2}>\ell_{t+1}$. This proves (A-$(t+1)$).

Thus we have defined the sequences $\{i_{t}\}$, $\{\ell_{t}\}$, closed connected oriented manifolds $\{P_{t}\}$ of dimensions $\{8\cdot i_{t}^{2}\}$ and homotopy equivalences $\{f_{t}\}$ of degree $1$ which satisfy the above properties.

Given a positive integer $d$, let

[TABLE]

and $p=\sum_{t=0}^{d}8\cdot i_{t}^{2}$. We show that $F_{t}\notin\mathfrak{h}_{\ell_{t}}(P)$ and $F_{t}\in\mathfrak{h}_{\ell_{t+1}}(P)$. Let $q_{t}:P\rightarrow P_{t}$ be the canonical projection. Then the stable tangent bundle $\tau_{P}$ is isomorphic to $q_{0}^{\ast}(\tau_{P_{0}})\oplus q_{1}^{\ast}(\tau_{P_{1}})\oplus\cdots\oplus q_{d}^{\ast}(\tau_{P_{d}})$. Hence, $\tau_{P}-F_{t}^{\ast}(\tau_{P})$ is equal to

[TABLE]

This shows that

[TABLE]

which does not vanish in $H^{2i_{t}^{2}}(P;\mathbb{Z})$ since $\mathbf{c}_{\mathbb{Z}}(\Sigma^{2i_{t}},\tau_{P_{t}}-f_{t}^{\ast}(\tau_{P_{t}}))\neq 0$ and since $q_{t}^{\ast}:H^{2i_{t}^{2}}(P_{t};\mathbb{Z})\rightarrow H^{2i_{t}^{2}}(P;\mathbb{Z})$ is injective. Furthermore, it follows from Proposition 4.3 that $\Sigma^{2i_{t}}(p,p)\subset W_{\ell+1}^{k}(p,p)$ and from Corollary 4.4 that $F_{t}$ is not homotopic to any $\mathcal{O}_{\ell_{t}}^{k}$-regular map. However, since $f_{t}$ is homotopic to an $\mathcal{O}_{\ell_{t+1}}^{k}$-regular map, $F_{t}$ is also homotopic to an $\mathcal{O}_{\ell_{t+1}}^{k}$-regular map. This proves the theorem.

We prepare further results which are necessary to study the filtration in (1.1). The assertions (i) and (ii) in the following theorem have been proved in [An2, Theorem 4.8] and [An4, Theorem 4.1] respectively, which are applications of the relative homotopy principles for $\mathcal{O}$-regular maps.

Theorem 5.3

Let $P$ be orientable and $f:P\rightarrow P$ be a smooth map.

(i) A map $f$ is homotopic to a fold-map if and only if $\tau_{P}$ is isomorphic to $f^{\ast}(\tau_{P})$.

(ii) If a map $f$ is $\Omega^{1}$-regular, then $f$ is homotopic to an $\Omega^{(1,1,0)}$-regular map.

Let $V(n,p)$ be an algebraic set of $J^{k}(n,p)$ which is invariant with respect to the actions of local diffeomorphisms of $(\mathbb{R}^{n},0)$ and $(\mathbb{R}^{n},0)$ and Let $V(N,P)$ be the subbundle of $J^{k}(N,P)$ associated to $V(n,p)$. By [B-H] we have the fundamental class of $V(N,P)$ under the coefficient group $\mathbb{Z}/2$, and have the Thom polynomial $\mathbf{c}(V(n,p),\tau_{N}-f^{\ast}(\tau_{P}))$ of $V(N,P)$.

Theorem 5.4

Let $V(p,p)$ be as above. Let $P$ be orientable and $f:P\rightarrow P$ be a smooth map.

(i) If $f$ is a homotopy equivalence, then $\mathbf{c}(V(p,p),\tau_{P}-f^{\ast}(\tau_{P}))$ vanishes.

(ii) $\mathbf{c}_{\mathbb{Z}}(W_{p}^{k}(p,p),\tau_{P}-f^{\ast}(\tau_{P}))=0$ for $p=5,6,7$ and

[TABLE]

for $p=8$.

(iii) Let $2\leqq p\leqq 8$. Then there exists a section $s$ of $\mathcal{O}_{p-1}^{k}(P,P)$ over $P$ with $\pi_{P}^{k}\circ s$ and $f$ being homotopic if and only if $\mathbf{c}_{\mathbb{Z}}(W_{p}^{k}(p,p),\tau_{P}-f^{\ast}(\tau_{P}))=0$.

Proof. (i) Let $S(\nu_{P})$ denote the spherical normal fiber space of $P$. It follows from [Sp] that $S(\nu_{P})$ is equivalent to $f^{\ast}(S(\nu_{P}))$. Hence, the associated spherical spaces of $\tau_{P}$ and $f^{\ast}(\tau_{P})$ are equivalent. In particular, the Stiefel-Whitney classes of $\tau_{P}-f^{\ast}(\tau_{P})$ vanish.

(ii) If $p\leqq 8$, then a map $f:P\rightarrow P$ is homotopic to a smooth map with only $\mathcal{K}$-simple singularities by [MaVI]. According to [F-R], the integer Thom polynomial of $W_{p}^{k}(p,p)$ is equal to the formula for $p=8$ and vanish for $p=5,6,7$ in $H^{p}(P;\mathbb{Z)\approx Z}$.

(iii) It follows from the relative homotopy principle for $\mathcal{O}_{p-1}^{k}$-regular maps $P\rightarrow P$ that the primary obstruction in $H^{p}(P;\pi_{p-1}(\mathcal{O}_{p-1}^{k}(p,p))$ is the unique obstruction for finding the required section. By an elementary argument we have

[TABLE]

This shows the assertion.

Finally we study the filtration in (1.1) in the case of $P$ being orientable and $p\leqq 8$ by applying the homotopy principles in Theorems 1.2 and 5.3. We have $\mathfrak{h}_{p}(P)=\mathfrak{h}(P)$.

Examples.

Case: $p\leqq 3$; $\mathfrak{h}_{0}(P)\subset\mathfrak{h}_{1}(P)=\mathfrak{h}(P)$.

Since $P$ is parallelizable, $TP$ and $f^{\ast}(TP)$ are trivial. So a map $f:P\rightarrow P$ is homotopic to a fold-map. We refer the reader to [Ru, 1].

Case: $p=4$; $\mathfrak{h}_{0}(P)\subset\mathfrak{h}_{1}(P)\subset\mathfrak{h}_{2}(P)=\mathfrak{h}_{3}(P)\subset\mathfrak{h}_{4}(P)$.

It is known that $\mathbf{c}_{\mathbb{Z}}(\Sigma^{4};\tau_{P}-f^{\ast}(\tau_{P}))=P_{2}(\tau_{P}-f^{\ast}(\tau_{P}))$. If this class vanish, then there exists a section $P\rightarrow\Omega^{1}(P,P)$ covering $f$, and hence an $\Omega^{1}$-regular map by [F]. By Theorems 5.3 and 5.4 we obtain an $\Omega^{(1,1,0)}$-regular map homotopic to $f$. It has been proved in [Ak] that $\mathfrak{h}_{0}(P)\neq\mathfrak{h}(P)$ for $P=S^{3}\times S^{1}\#S^{2}\times S^{2}$.

Case: $5\leqq p\leqq 7$; $\mathfrak{h}_{0}(P)\subset\mathfrak{h}_{1}(P)\subset\cdots\subset\mathfrak{h}_{p-1}(P)=\mathfrak{h}_{p}(P)$.

This follows from Theorems 1.2 and 5.4.

Case: $p=8$; $\mathfrak{h}_{0}(P)\subset\mathfrak{h}_{1}(P)\subset\cdots\subset\mathfrak{h}_{7}(P)\subset\mathfrak{h}_{8}(P)$.

If $9P_{2}(\tau_{P}-f^{\ast}(\tau_{P}))+3P_{1}^{2}(\tau_{P}-f^{\ast}(\tau_{P}))=0$, then the homotopy class of $f$ lies in $\mathfrak{h}_{7}(P)$ by Theorems 1.2 and 5.4.

For more precise information we must investigate the obstructions for finding sections in $\Gamma_{\mathcal{O}_{\ell}^{k}}(P,P)$ related to $W_{\ell+1}^{k}(p,p)$.