$6\mathrm{V}\mathrm{I}\mathrm{I}\mathrm{I}$ (p (Kazuhiro Sakuma) Dept. of Math. and Phys., Kinki Univ.,. (,,.) \S 0. $C^{\infty

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1 $6\mathrm{V}\mathrm{I}\mathrm{I}\mathrm{I}$ (p (Kazuhiro Sakuma) Dept of Math and Phys Kinki Univ ( ) \S 0 $M^{n}$ $N^{p}$ $n$ $p$ $f$ $M^{n}arrow N^{p}$ $n<p$ $N^{p}=\mathrm{R}^{p}$ $M^{n}arrow \mathrm{r}^{p}$ $f$ ( )? 1930 Whitney ([1] )? Chern Thom [ ] ( 3 ) 1370)

2 112 ( ) $n>p$ $p=1$ ( ) $f$?? $n<p$ $n\geq p$ ( ) $N^{p}=\mathrm{R}^{p}$ $f$? ( $M^{n}arrow \mathrm{r}^{p}$ $f$ ) ( ) 1 \S 1 $n\geq p$ $f$ $M^{n}arrow \mathrm{r}^{p}$ [ $S(f)=$ { $x\in M^{n}$ ;rank $df_{x}<p$ } $f$ $x\in S(f)$ $(x_{1} \ldots x_{n})$ $x$ $(y_{1} \ldots y_{p})$ $f(x)$ $y_{i}\mathrm{o}f=x(1\leq i\leq p-1)$ $y_{p}\mathrm{o}f=\pm x_{p}^{2}\pm\cdots\pm x_{n}^{2}$ $f$ $f$ $M^{n}arrow \mathrm{r}^{p}$ ( $p=1$ ) $f$ $M^{n}arrow \mathrm{r}^{p}$?? 1 counterpart

$113 $p$ $p\geq 2$ $p$ $p=2$ 11 (R Thom H Levine) \mathrm{r}^{2}$ $f$ $M^{n}arrow\cdot Levine ( $n\geq 3$ $n=2$ ) Thom stratffication strata Thom ) $\Rightarrow-$ Stiefel-Whitney $w_{n}\in$

3 113 $p$ $p\geq 2$ $p$ $p=2$ 11 (R Thom H Levine) \mathrm{r}^{2}$ $f$ $M^{n}arrow\cdot Levine ( $n\geq 3$ $n=2$ ) Thom stratffication strata Thom ) $\Rightarrow-$ Stiefel-Whitney $w_{n}\in H^{n}(M^{n}; \mathrm{z}_{2})$ $p\geq 3$ $n\geq p$ $p$ Y Eliashberg $M^{n}arrow \mathrm{r}^{p}$ stably parallelizable $f$ ([2] ) ( ) stably parallelizable ( [3] ) -\acute \supset $S^{2}$ $S^{2}$ $S^{2}\cross S^{2}\sim$ $\mathrm{a}\mathrm{a}$ ( 2Stiefel-Whitney $w_{2}(s^{2}\cross S^{2})\sim\neq 0$ ( stably parallelizable $\mathrm{a}\mathrm{a}$) $f$ $S^{2}\cross S^{2}\simarrow \mathrm{r}^{3}$ $S^{2}\cross S^{2}\sim$ 2] (Kikuchi-Saeki[4] Saeki-Sakuma[5]) $n$ $M^{n}arrow \mathrm{r}^{p}$ $f$ $p$ 13 7 $p=1$ $p=37$ $\mathrm{r}^{p}(p=37)$ $p=37$ [5]

$114 1 ( ) ( J F Adams ) ( 2) ( ) $p=37$ ( 2 ) 3] (Saeki [6]) $\mathrm{c}p^{2}$ 4 ( ) $M^{4}arrow \mathrm{r}^{3}$ $f$ N $C^{\infty}(M^{n}$ $S$ $S$ ( ) 3 $M^{4}arrow \mathrm{r}^{3}$ $f$ (1)$

4 114 1 ( ) ( J F Adams ) ( 2) ( ) $p=37$ ( 2 ) 3] (Saeki [6]) $\mathrm{c}p^{2}$ 4 ( ) $M^{4}arrow \mathrm{r}^{3}$ $f$ N $C^{\infty}(M^{n}$ $S$ $S$ ( ) 3 $M^{4}arrow \mathrm{r}^{3}$ $f$ (1) $y\mathrm{o}f=x\cdot(i=12)$ $y_{3}\mathrm{o}f=x_{3}^{2}\pm x_{4}^{2}$ (2) $y_{i}\mathrm{o}f=x\cdot(i=12)$ $y_{3}\mathrm{o}f=x_{3}^{3}+x_{1}x_{2}\pm x_{4}^{2}$ (3) $y\mathrm{o}f=x_{i}(i=12)$ $y_{3}\mathrm{o}f=x_{3}^{4}+x_{1}^{2}x_{2}+x_{1}x_{3}\pm x_{4}^{2}$ (1) (2) (cusp) (3) (swallow tail) (1) 2 (2) 1 (3) ( [1] ) $A_{1}(f)$ A2(f) $A_{3}(f)$ $S(f)=\overline{A_{1}(f)}=A_{1}(f)\cup A_{2}(f)$ $\cup A_{3}(f)$ A2(f) $=A_{2}(f)$ $\cup A_{3}(f)$ $\overline{x}$ ( $X$ ) $\mathrm{m}\mathrm{o}\mathrm{d} 2$ 2 ( ) 3 ([1] )

$115 Thom 4 $[S(f)]_{2}^{*}=w_{2}\in H^{2}(M^{4};\mathrm{Z}_{2})$ $[\overline{a_{1}(f)}]_{2}^{*}=0$ $[A_{3}(f)]_{2}^{*}=0$ ( $*$ ) Thom 0 4 ( ) 5 80 $\Re\backslash$ 6 3 Thom Thom?$

5 115 Thom 4 $[S(f)]_{2}^{*}=w_{2}\in H^{2}(M^{4};\mathrm{Z}_{2})$ $[\overline{a_{1}(f)}]_{2}^{*}=0$ $[A_{3}(f)]_{2}^{*}=0$ ( $*$ ) Thom 0 4 ( ) 5 80 $\Re\backslash$ 6 3 Thom Thom? $\mathrm{o}\mathrm{r}\mathrm{y}$ Dynamical Arnol d Vassiliev Goryunov and Lyashko Singularities Local and Global the $\mathrm{i}\mathrm{v}$ System Encyclopeadia Math vol 6 Springer-Verlag 1993 Chapter 4 \S 5 $\text{ }$ ( Vassiliev ) $\mathrm{m}\mathrm{o}\mathrm{d} 4$ ( ) 4 Thom $f$ $M^{n}arrow N^{p}$ t e Stiefel-Whitney $w_{1}(m^{n})$ $f^{*}wj(tn^{p})$ Chern Thom Rim\ anyi $n<p$ Thom ( ) Richard Rim\ anyi Invent math 143 (2001) $n\geq p$ Thom $n\geq p$ 5 4 ( ) [1] 5 $6\mathrm{Y}\mathrm{A}\mathrm{n}\mathrm{d}\mathrm{o}$ On the $471\triangleleft 87$ elimination of Morin singularities J Math Soc Japan 37 (1985) $=\mathrm{n}d$

6 116 4] 4 $H(M^{4};\mathrm{Z})\ovalbox{\tt\small REJECT} 0$ $f\ovalbox{\tt\small REJECT} M^{4}arrow \mathrm{r}^{3}$ $\sigma(m^{4})\equiv-s(f)\cdot S(f)$ $(\mathrm{m}\mathrm{o}\mathrm{d} 4)$ (11) $\sigma(m^{4})$ (signature) $S(f)\cdot S(f)$ ( 2 ) 1 4 (11) $S(f)\cdot S(f)\equiv-\sigma(M^{4})\equiv 3\sigma(M^{4})(=p_{1}[M^{4}])$ $(\mathrm{m}\mathrm{o}\mathrm{d} 4)$ (12) Hirzebruch $p_{1}[m^{4}]$ Pontrjagin 4 1 (12) $p_{1}[m^{4}]=s(f)\cdot S(f)$ (13) (!) \S 2 $n$ $f$ $M^{n}arrow \mathrm{r}^{p}$ $S(f)$ $S(f)$ ( ) $(p-1)$ $S(\cdot f)$ Mn ( $n$ ) $f$ $S(f)$ $S(f)$ $\tilde{s}$ $2(p-1)-n=2p-n-2$ $\tilde{s}$ $\mathrm{z}$ $[\tilde{s}]\in H_{2p-n-2}(M^{n};\mathrm{Z})$ $[\tilde{s}]^{*}\in H^{2(n-p+1)}$ ( ; Z) $I(S(f))$ $f$ (self-intersection class)

7 117 (12) $f$ $M^{n}arrow N^{p}$ $n$ $p$ $n\geq p$ $x\in S(f)$ [ $x_{n}\rangle$ ( $x_{1}$ $x$ $\ldots$ $f(x)$ $(y_{1} \ldots y_{p})$ $y_{i}\circ f=x_{i}(1\leq i\leq p-1)$ $y_{p} \mathrm{o}f=x_{p}^{l+1}+\sum_{i=1}^{l-1}x_{i}x_{p}^{l-}\pm x_{p+1}^{2}\pm\cdots\pm x_{n}^{2}$ A $l=1$ $l=2$ $l=3$ $f$ $f$ ([5] ) $S(f)$ $(np)=(43)$ 5] $(\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{o}\mathrm{t}+\mathrm{s}\mathrm{a}\mathrm{e}\mathrm{k}\mathrm{i}- $n$ $f$ $M^{n}arrow \mathrm{r}^{p}$ $n-p+1=2k(n\geq$ \mathrm{s}\mathrm{a}\mathrm{k}\mathrm{u}\mathrm{m}\mathrm{a}[7])^{7}m^{n}$ $n$ (2 $p\geq 1$ $k\geq 1)$ $I(S(f))=p_{k}(TM^{n})\in H^{4k}(M^{n};\mathrm{Z})$ modulo 4-torsion $p_{k}(tm^{n})$ $k$ $n=4$ $p=3$ 4 torsion (12) integral version ( integral formula 3 ) $f$ $\mathrm{c}p^{2}\#\mathrm{c}p^{2}arrow \mathrm{r}^{3}$ $ \mathrm{h}\yen 7 7\grave{J}rightarrow$\Xi \ddagger

8 118 $f\ovalbox{\tt\small REJECT} \mathrm{c}p^{2}\#\mathrm{c}p^{2}arrow \mathrm{r}^{3}$ 3 $y_{3}\mathrm{o}f=x_{3}^{2}+x_{4}^{2}$ $of=x_{3}^{2}-x_{4}^{2}$ $y_{3}$ ; $S(f)=S^{+}(f)\cup$ $S^{+}(f)$ $S^{-}(f)$ Saeki[6] (i) $S^{+}(f)$ (ii) $\beta\in H_{2}(\mathrm{C}P^{2}\#\mathrm{C}P^{2}; \mathrm{z})\cong \mathrm{z}\oplus \mathrm{z}$ $\alpha$ (i) $S^{+}(f)$ { $[S^{+}(f)]=p\alpha+q\beta(p q\in \mathrm{z})$ $\sigma(\mathrm{c}p^{2}\#\mathrm{c}p^{2})=2$ integral formula $S^{-}(f)$ $6=p_{1}[\mathrm{C}P^{2}\#\mathrm{C}P^{2}]=S(f)\cdot S(f)=S^{+}(f)\cdot S^{+}(f)+S^{-}(f)\cdot S^{-}(f)$ (ii) I $S^{-}(f)\cdot S^{-}(f)=0$ $6=[S^{+}(f)]\cdot[S^{+}(f)]=(p\alpha+q\beta)^{2}=p^{2}+q^{2}$ $S^{2}\cross S^{2}\sim$ $\mathrm{c}p^{2}\#\overline{\mathrm{c}p^{2}}arrow \mathrm{r}^{3}$ $f$ $\mathrm{r}^{3}$ [ ] $\mathrm{r}^{3}$ preprint 5 [ 6] ([7]) $f$ $M^{n}arrow \mathrm{r}^{p}$ $n-p+1=2k$ $x\cup x=p_{k}(tm^{n})$ modulo 8-torision $x\in H^{2k}(M^{n};\mathrm{Z})$ 4 2 ( ) $\mathrm{r}^{7}$ $\mathrm{a}\mathrm{a}$ 8

9 $M^{8}\ovalbox{\tt\small REJECT} \mathrm{c}p^{4}$ $\iota_{!}$ 119 $\mathrm{h}p^{2}$ $f\ovalbox{\tt\small REJECT} M^{8}arrow \mathrm{r}^{7}$ $\alpha^{2}\in H^{4}(\mathrm{C}P^{4}\ovalbox{\tt\small REJECT} \mathrm{z})\ovalbox{\tt\small REJECT} \mathrm{z}$ $5\alpha^{2}\neq 0$ $u\in H^{4}(\mathrm{H}P^{2}\ovalbox{\tt\small REJECT} \mathrm{z})\ovalbox{\tt\small REJECT} \mathrm{z}$ 6 $x\mathrm{c}h^{2}(\mathrm{h}p^{2};\mathrm{z})$ $p (\mathrm{c}p^{4})\ovalbox{\tt\small REJECT}$ $p_{1}(\mathrm{h}p^{2})\ovalbox{\tt\small REJECT} 2u\neq 0$ $x\cup x=p_{1}(\mathrm{h}p^{2})=2u\neq 0$ $H^{2}(\mathrm{H}P^{2}; \mathrm{z})=0$ $y\in H^{2}(\mathrm{C}P^{4};\mathrm{Z})$ 6 $y\cup y=p_{1}(\mathrm{c}p^{4})=5\alpha^{2}\neq 0$ $H^{2}(\mathrm{C}P^{4};\mathrm{Z})\cong \mathrm{z}$ $\alpha$ $k\in \mathrm{z}$ $y=k\alpha$ $k^{2}=5$ $\mathrm{c}p^{4}$ $\mathrm{h}p^{2}$ 8 $M^{8}=\#{}_{k}\mathrm{C}P^{4}\#\iota\overline{\mathrm{C}P^{4}}$ $l$ $\#{}_{k}\mathrm{h}p^{2}\#\iota\overline{\mathrm{h}p^{2}}$ $k$ } $k+l\geq 1$ $M^{8}=\mathrm{C}P^{2}\cross \mathrm{c}p^{2}$ $\mathrm{c}p^{4}$ $\mathrm{c}p^{2}\cross \mathrm{c}p^{2}$ $\Omega^{8}\cong \mathrm{z}\oplus \mathrm{z}$ 8 oriented cobordism group \S [ desingularization method { [ $p=$ $n-1$ ( $k=1$ ) $x\in M^{n}$ $TM_{x}^{n}$ $\pi$ 2 $Garrow M^{n}$ $G$ tautological $\gamma$ 2-plane bundle $\pi^{*}f^{*}t\mathrm{r}^{n-1}=\epsilon^{n-1}$ $G$ $\mathrm{h}\mathrm{o}\mathrm{m}(\gamma \epsilon^{n-1})$ $s$ $s$ preimage $s^{-1}(0)$ $\tilde{s}(f)$ $j$ $\tilde{s}(f)\mapsto G$ { orientation double cover I $\mathrm{a}$ $\iota$ $\nu$ $S(f)\mathrm{c}arrow M^{n}$ $\tilde{\pi}$ $\tilde{\pi}$ $\tilde{s}(f)arrow S(f)$ Gysin $H^{*}(S(f);\mathcal{Z})arrow H^{*+2}(M^{n};\mathrm{Z})$

10 120 $S(\ovalbox{\tt\small REJECT}$ nonorietable $\mathcal{z}$ $\ovalbox{\tt\small $\in H^{2}(S(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} Z)$ REJECT}\nu$) $\nu$ twisted Euler $I(S(f))=$ $(e(\nu))\in H^{4}(M^{n}; \mathrm{z})$ $S(f)$ (selfintersection class) [ \S 2 $j_{*}[\tilde{s}(f)]^{*}=e(\mathrm{h}\mathrm{o}\mathrm{m}(\gamma\epsilon^{p}))=(e(\gamma))^{p}$ $S(f)$ $K=$ $\mathrm{k}\mathrm{e}\mathrm{r}df$ $Q=\mathrm{C}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}df$ $\mathrm{h}\mathrm{o}\mathrm{m}(k Q)\cong\nu$ $\tilde{s}(f)$ $\mathrm{h}\mathrm{o}\mathrm{m}(j^{*}\gamma \epsilon^{p}/{\rm Im} df)\cong\tilde{\pi}^{*}\nu$ $e(j^{*}\gamma)=e(\mathrm{h}\mathrm{o}\mathrm{m}(j^{*}\gamma\epsilon^{p}/{\rm Im} df))=e(\tilde{\pi}^{*}\nu)$ $\pi_{!}((e(\gamma))^{n-2})=2$ \pi *TM=\gamma \oplus \gamma $k \leq N=[\frac{n-2}{2}]$ $\pi^{*}p_{k+1}(tm)-p_{k+1}(\phi^{[perp]})=p_{1}(\phi)p_{k}(\phi^{[perp]})$ modulo 2-torsion $j_{!}(e(j^{*}\phi))$ $=$ $e(\phi)j_{!}(1)=e(\phi)^{n}=p_{1}(\phi)e(\phi)^{n-2}$ $=$ $(\pi^{*}p_{1}(tm)-p_{1}(\phi^{[perp]}))e(\phi)^{n-2}$ $=$ $\pi^{*}p_{1}(tm)e(\phi)^{n-2}-p_{1}(\phi^{[perp]})p_{1}(\phi)e(\phi)^{n-4}$ $=$ $\pi^{*}p_{1}(tm)e(\phi)^{n-2}-(\pi^{*}p_{2}(tm)-p_{2}(\phi^{[perp]}))e(\phi)^{n-4}$ $=$ $\pi^{*}p_{1}(tm)e(\phi)^{n-2}-\cdots\pm\pi^{*}p_{n+1}(tm)e(\phi)^{n-2n}$ adjunction formula $\pi_{!}j_{!}(e(j^{*}\phi))$ $=p_{1}(tm)\pi_{!}(e(\phi)^{n-2})-p_{2}(tm)\pi_{!}(e(\phi)^{n-4})+\cdots$ $=$ $2p_{1}(TM)$

11 121 $-\text{ }$ $i_{!}\tilde{\pi}_{!}(e(j^{*}\phi))=i_{!}\tilde{\pi} (e(\tilde{\pi}^{*}\nu))=i_{!}((e(\nu)\tilde{\pi}_{!}(1))=i_{!}(2e(\nu))=2i_{!}(e(\nu))$ $i_{!}\tilde{\pi}_{!}$ $=\pi j $ $2i_{!}((e(\nu))=2p_{1}(TM)$ $2k$ tautological bundle $2k$ $\mathrm{a}$ $\mathrm{a}$ 2 $2k$ [1] ( ) [2] $\mathrm{j}$ $\mathrm{m}$ \ E $\mathrm{h}\mathrm{a}\check{\mathrm{s}}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$ Surgery of singularities of smooth mapp $ir$ $gs$ Math USSR Izv 6(1972) $\mathrm{r}^{3}$ [3] 0 Saeki and K Sakuma On special generic maps into Pacific J Math 184(1998) [4] S Kikuchi and 0 Saeki Remarks on the topology of folds Proc Amer Math Soc 123(1995) [5] 0 Saeki and K Sakuma Maps with only Motin singularities and the Hopf invaria $nt$ one proble $m$ Math Proc Camb Phil Soc 124 (1998) [6] 0 Saeki Note on the topology offolds J Math Soc Japan (1992) [7] T Ohmoto 0 Saeki and K Sakuma The self-intersection class and nonexiste $nce$ of fold maps preprint

42 1 ( ) 7 ( ) $\mathrm{s}17$ $-\supset$ 2 $(1610?\sim 1624)$ 8 (1622) (3 ), 4 (1627?) 5 (1628) ( ) 6 (1629) ( ) 8 (1631) (2 ) $\text{ }$ ( ) $\text{

$42 1 ( ) 7 ( ) $\mathrm{s}17$ $-\supset$ 2 $(1610?\sim 1624)$ 8 (1622) (3 ), 4 (1627?) 5 (1628) ( ) 6 (1629) ( ) 8 (1631) (2 ) $\text{ }$ ( ) $\text{$ 26 [\copyright 0 $\perp$ $\perp$ 1064 1998 41-62 41 REJECT}$ $=\underline{\not\equiv!}\xi*$ $\iota_{arrow}^{-}\approx 1,$ $\ovalbox{\tt\small ffl $\mathrm{y}