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σn = (α0, . . . , αn)gi`c\

x =

n∑

j=0

xjαj , ∀j xj ≥ 0 ,

n∑

j=0

xj = 1 .

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σ(n) k;h X � σ t�bg³bdwa��|zkpqnma~u\±���=´²Za\zV7��\lq(wab�X*m]`ck��p\¸X[Z�g�X:t�hτ (q) t�b=g³|z\,~u~>k;h X giVae τ (q)∩σ(n) 6= ∅X[Za\zV

τ (q) ⊂ σ(n) �"vxtuV]|,\ σ(n) ⊂ Xn−1 �¸��\�>Vak�� X[Z�giX τ (q) ∩ σ(n) 6= ∅�lXcZa\,V

q ≤ n− 1�A�F\Kma`ck��p\�XcZa\Kma`ckpm8kpbdt�X[t�kpV ��� ea\zbc|,\zVaeat�Vao�t�Vaeawa|OX[tuk;V kpV

q�

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Xt�b�g_`d\,opw]~�g;`³¶�� |,kpqnma~u\±�x\;�

σ(n) t�b�Zakpqn\,kpqnkp`cm]Zatu|³X[kyg;V(n − 1)

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τ��´²Z>wab

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τ ∩ σ(n) 6= ∅��\{q(wab�X�Zag��;\

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�lg;Vae£bcwamam8kpbd\XcZ�giXτ (q) ⊂ Xq − C

�£vxt�Va|,\τ (q) t�bµkpm8\,V£tuV Xq

�τ (q) tub_kpm8\,V£tuV Xq − C

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Xq ∪C�����µZ��Dm�kiX[Za\zbctubz�

σ(n) ⊂ C ∪Xq�>bckKX[ZagiX

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�τ (q) tub�Zakpqn\,kpqnkp`cm]Zatu|³X[k Rd

��g;Vaeσ(n) t�b�Zakpqn\,k;qykp`dmaZat�|BX[k

Sn−1 �

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q6W|z\,~u~*k;h

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M�>��t�X[Z

Kp

X[Za\�|z\,~u~�bAk;heatuqn\,VabdtukpV

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p+ 1\zXh : B →M

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bCcedgf�hjikhml+fonqp 2n�� ,;°σ(p) §u¨L,`c\zopwa~ug;`�hCg;|z\®�{ τ (p+1) § {

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htub(g�Zakpqn\,kpqnkp`dmaZatubdq g;V

σ ⊂ τ��\ op\zX

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g;Vaehtubng

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τ�

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ceν ν.

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τ (p+r) > ν(p−1) ��1�¯�«w.Y¬j¯�«Oª[«�,;ª[«7¬���® (p + r − 1) v ©}«w/j/ ¨ σ(p+r−1),0.8¦σ(p+r−1) ¨w*�©}¯¬j¯�,;¬ σ 6= σ

,0.8¦τ > σ > ν, τ > σ > ν.

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�8��������� ����� � ���� ��  �

¥:tuopwa`d\ � � � � � �l¶�kp~u~ug;mabct�Vaoµg26 bct�qyma~�\O��kpV�X[kngµ�p\,`�X[\±�

¥:tuopwa`d\ � � � � � �*¶�kp~u~ug;mabct�Vao�X[Za\²�8kpwaVaeag;`d�fk;h3X[Za\�|,wa�8\²�aw]XAk;Va\�hCg;|,\i�pkpV�X[kKg��;\,`�X[\O�

� klql �]´²Z]\�ma`ckDk;h]tubM���ft�Vaeawa|OX[t�kpVfk;Vr�Avxwam]m�kpbd\

r = 1�0X[Z�g�X*t�b

τ (p+1) > σ(p−1) �vDtuVa|z\Mtub�`c\,o;wa~�g;`z�DXcZa\

p6W|,\z~u~�b�t�V

τ − τg;`c\�ea\,Vabd\�tuV

τ − τ�s`c\zq�g;`d� % � � � � �±�xt.� \i�

σ(p)<τ

σ = τ − τ.

´²Z>wab{X[Za\z`c\n\O�xtub�X[bfgp6W|z\,~u~

σ(p) ��t�XcZ τ > σbdwa|}Z¡X[ZagiX

σ > ν��´²Z]\,kp`d\,q % � � �

o;w�g;`[giV>Xc\,\zb�XcZa\K\O�xtub�X[\zVa|,\{k;hσ(p) 6= σ

bdwa|}Z$XcZ�giXτ > σ > ν

�¥�kp`�op\,Va\z`[g;~

r����\ng;o�g;t�V Z�g��p\µX[Za\

(p + r − 1)6W|z\,~�~ub�e]\,Vabd\yt�V

τ − τ�(´²Z>wabK��\

|,g;V$^aVae�g(p + r − 1)

6 |,\z~u~σ��t�X[Z

τ > σ > ν�²¶�kpV�X[tuV>wat�VaontuVTX[Zat�b�hCgibcZat�kpV��a��\

|,g;VY^�VaeÀg(p + r − 2)

6W|z\,~�~νbdwa|}Z¡XcZ�giX

σ > ν > ν� � mama~��xt�VaoTX[Za\zkp`c\zq % � � �

Xck�X[Z]\fX[`dtuma~�\τ > σ > ν

��\(~�\�g;`dV�X[Za\z`c\ftubQg(p + r − 1)

6 |,\z~u~σ 6= σ

bdwa|}Z�X[Z�g�Xτ > σ > ν

�¸´²Za\,V��3XcZa\K|,\z~u~ubσg;V]e

σb[g�X[tub�hj�#X[Z]\{e]\,bct�`c\ze$ma`dkpm8\,`dXctu\zb,�

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 � ��������� ������������������������������ ���������!�#"%$

9;:�� � LzS�D ? ��J � H IL?*S � J�� �²IL?�N

O�P���PSQ �uU�a�[ � _ !8_��I^ � a�_ � � ��[!�U�^(�

UWVÀX[Zat�b(bc\z|zX[t�kpV�� ��\ tuV�X[`dkxe]wa|,\�XcZa\#qyg;tuVFea\z^aVat�XctukpVabz�!�M\OXM�8\Tg$^�Vat�X[\T¶��

|,kpqnma~u\±�x\;�bCcedgf�hjikhml+fonqp ��t�eatubd|,`c\OX[\(�$kp`cbd\7hswaVa|OX[t�kpV�®(.

M§u¨%, { *�.�©O¬C§G®(.

f : K → R¨�,;¬.§u¨m{±°�§j.�4�{�®�ªG,0/j/σ ∈ Kp

| §�}/ m{σ§u¨G,0. §jª±ª}«�40*�/7,;ª�{=,D©}«µ®�{

τ (p+1) ¬s¯�«w. f(τ) > f(σ)�³­¼®�ª[«}®(y;«Oª �

#{τ (p+1) > σ|f(τ) ≤ f(σ)} ≤ 1.

| §j§�} �3§ν(p−1) §u¨�,0.�§jª±ª[«�40*�/7,;ª {=,D©}«7®�{ σ ¬s¯�«w. f(ν) < f(σ)

�³­¼®�ª[«}®(y;«Oª �#{ν(p−1) < σ|f(ν) ≥ f(σ)} ≤ 1.

bCcedgf�hjikhml+fonqp "��M«O¬f,T¦;§u¨�©Oª[«O¬ «�­¼®�ª[¨�« { *8.�©O¬C§G®(.¡®(.

M� -n«�¨�,;°

σ ∈ Kp

§u¨,T|z`ct�Xctu|,g;~�m8kptuV�X�k;h:tuV]ea\O�#mÀ§ {| §�}

#{τ (p+1) > σ|f(τ) ≤ f(σ)} = 0.

| §j§�}#{ν(p−1) < σ|f(ν) ≥ f(σ)} = 0.

��x ,0'�)+/u«��l¶�kpVabdtuea\z`²X[Za\f¶�� |zkpqnma~u\±� �8\,~uk�� �

¥*tuopw]`c\ � � � �  �A´M��kyeatubd|,`c\OX[\�hswaVa|OX[t�kpVab²kpV$gn¶�� |,k;qyma~�\O�

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�8����� ��� �q�������u� ������ ���������$  ' V�kpVa\fZ�g;Vae�����\(bc\z\7X[Z�g�XBXcZa\7hswaVa|OX[t�kpV�t�V$^�opw]`c\ % � � �  �stj�²tub³Vak;XQgyeat�bc|,`d\zXc\

�kp`cbd\³hswaV]|zX[t�kpV�� �$kp`c\�ma`d\,|ztubc\z~��(XcZa\³\,eaop\f−1(0)

�Dtuk;~�giXc\,b `dwa~u\G�Ct�tj�:kih�ea\O^�Vat�X[tuk;V% � � D�3g;Vae$XcZa\K�p\,`�X[\±�

f−1(3)�Dt�kp~�giXc\,b�`cw]~u\��Ct ��k;h ea\O^�Vat�X[tuk;V % � � x� ' V$XcZa\fkiX[Za\z`

Zag;Vae�� X[Za\�hswaV]|zX[t�kpVTk;h=^�opwa`d\ % � � �  �Ct�tj��t�b�g_e]tubc|z`c\OX[\f�$kp`dbc\KhswaVa|zXctukpV���gx ,0'�)+/u«��:e]\z^�Vat�X[t�kpVab % � � ³giVae % � � � tuqnma~��BXcZ�giX=t�h

Mtub=g�`c\zopwa~�gi`=¶�� |,kpqnma~�\O�x\;�

XcZa\,VnX[Za\�qntuVat�q(waq�k;hfq(w]bdXlkD|,|,w]`lVyg��p\,`�X[\±���D��Zatu|}Znq(wab�Xl��\³gK|,`ct�X[t�|�g;~am8kpt�V>X

kih*t�Vaea\O�0�

UWV]ea\,\ze���t�hp ≥ 1

�xX[Za\zV\,g;|}Zp6W|z\,~u~8Z�g;b�g�X�~u\�gibdX �

(p− 1)6GhCg;|,\zbL�C|zkpVabc\z½>wa\,Va|z\fkih

XcZa\,k;`c\,q % � � �e�O�U X³hskp~u~�k���b�hs`ckpq e]\z^�Vat�X[t�kpV % � � � XcZ�giXBg

p6W|z\,~�~

σt�b³V]k;X³|,`ct�X[t�|�g;~�t�h giVae$k;Va~�� t�h

\zt�XcZa\,`²k;h=X[Z]\Khsk;~u~uk���t�Vao(|,k;Vaeat�XctukpV]b²Zakp~ue]b�Ct �

∃τ (p+1) > σbcwa|}ZX[Z�giX

f(τ) ≤ f(σ)�

�stut �∃ν(p−1) < σ

bdwa|}Z$XcZ�giXf(ν) ≥ f(σ)

�� c���� nqp % &�®(.8¦;§j¬C§G®(.]¨ | §�}�,0.8¦ | §j§�} ©5,0.z.�®�¬��O®�¬j¯��O«{¬Cª *3«��� klql �A¶�k;Vaeat�XctukpV��stut ��`c\z½Dw]tu`c\zb

p ≥ 1��Z]tu|}ZT��\{Vak���g;bdbcwaqn\;�

vDwamam8kpbc\ �Ct �7tubµX[`dwa\;� ´²Za\,Vσq(w]bdXy��\gY`c\,o;wa~�g;`(hCgi|,\k;h

τ�CbdtuVa|z\

f`d\,bcm8\,|OX

mag;`dXct�g;~¸kp`dea\,`(k;VÀtu`d`c\,o;wa~�g;`�hCg;|z\,b �O�Y¥�`ckpq�|zkpVaeat�X[tuk;Vo�stj�{kih�e����[^aVat�XctukpV % � � x� t�hσ 6= σ

t�b�g;Vak;XcZa\,`p6GhCg;|,\Kk;h

τ�a��\Kq(wab�XBZag��;\

f(σ) < f(τ).� � � � � Â �

´²Z]\,`c\Ohskp`c\f(σ) < f(σ)$³k��5bcw]mam�k;bc\{|,kpVae]t�X[t�kpV��Ct�tj�¸tub�X[`dwa\;�¸´²Za\zV

νq(w]bdXBg;~ubdkµ��\{gµ`c\zopwa~ug;`�hCg;|z\{kih

σ�

�l�#X[Za\zkp`c\zq % � � � �]XcZa\,`d\f\±�xtubdXcb³gp6W|z\,~u~

σ 6= σbdwa|}Z$XcZ�giX

τ > σ > ν�

¥�`dkpq |,kpV]eat�XctukpV��Ct�tj�(k;h�e����[^�Vat�X[t�kpV % � � D�f(ν)

|,g;VaVakiX��8\≥��kiX[Z

f(σ)g;Vae

f(σ)�A´²Z>wab

f(ν) < f(σ)�l¶�kpq7�atuVat�Vao(X[Z]tub���t�XcZ � % � � �  �l��\{kp�]X[g;tuV

f(σ) ≤ f(ν) < f(σ) < f(τ) ≤ f(σ)��Z]tu|}Ztub�gµ|,kpV�Xc`[g;eat�|zXctukpV��

OVP���P-O �I^ � a�_ !���_@^ ��_�W a �3^ � � _ � ��ZR\ � ��� [2^!WYXZ�_2`�_@aUWV¢XcZatub�bc\,|OX[t�kpV��²��\���t�~u~�ea\zqykpV]bdX[`cgiX[\ �$kp`cbd\�X[Za\zkp`c\zqyb�tuV�XcZa\�|�gibc\ k;h�^�V]t�X[\`d\,opw]~�g;`�¶�� |zkpqym]~u\O�x\zb,� � ~�~3g;~uk;Vao�X[Zat�blbd\,|OX[tuk;V��

M��tu~�~���\BgK^aVat�Xc\B`d\,opwa~ug;`l¶��

|zkpqnma~u\±�Tg;Vaefgµeatubd|,`c\OX[\f�$kp`dbc\KhswaVa|OX[t�kpVTkpV

M�

b�cedgf�hjikhml f nqp� � �=®�ªc ∈ R � ��«7¦D«m� .�«

M(c) =⋃

σ∈K

f(σ)≤c

τ≤σ

τ.

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 � ��������� ������������������������������ ���������!�#"%$

´²Z�g�X�tubM(c)

tublX[Za\�bd\zX�k;h=g;~�~�|z\,~�~ub�k;V#��Zatu|}Zft�b≤ c��g;b���\,~�~�g;b�gi~u~8k;h�X[Za\ztu`

hCg;|,\zb,�¸UWV$mag;`dXctu|,w]~�g;`Mtub�gµbdwa��|zkpqnma~u\±�Tk;h

M�

ºB\,`d\ft�b�gn~u\zqyqyg(��Zatu|}Z$opt��;\,b�w]b�gn|,`ct�X[\z`ct�g(Xck�bc\z\7t�hσbdwa|}Z�X[ZagiX

f(σ) > c~utu\zb

tuVM(c)

�A´=kµbc\,\�XcZ�giX��xt�X�bcw��n|,\zb²X[k7^�V]eτ��t�XcZ

σ < τg;Vae

f(τ) ≤ cg;Vae qnkp`d\

ma`c\z|,tubd\,~���X[Z]\{~�\,qnq�g7Xc\,~�~ub�wab²Xckn|zkpVabdtuea\z`³k;Va~��τ��t�X[Z

dim τ = dim σ + 1�

�!c���� nqp��@r �=«O¬σ�O«%,

p v ©}«w/j/*®�{ M ,0.8¦y¨w* )e)�®�¨�«τ > σ

�C1�¯�«w. ¬s¯�«Oª[«_«Sxp§u¨O¬s¨,

(p+ 1) v ©}«w/j/ τ �A§j¬s¯σ < τ ≤ τ

,0.8¦f(τ) ≤ f(τ)

�� klql �=�¡\Kma`dk0�;\{XcZatub²��� t�Vaeawa|OX[t�kpVTkpV X[Za\Keat�qy\zVabct�kpV k;h

τ�

vxtuVa|z\τ > σ

�=��\�Z�g��;\dim τ > dim σ

�#U hdim τ = p + 1

�M��\�|�giV¡|,k;Vabct�ea\,`τ = τ

�vxwamam8kpbc\

dim τ = p + r, r > 1�>�_XcZa\,kp`d\,q % � � �]��X[Z]\,`c\B\O�xtub�X[b(p+ r− 1)

6W|,\z~u~�bν1

giVaeν2

b[giXctub�hj�xt�Vaoτ > ν1 > σg;Vae

τ > ν2 > σ�¸¶�kpV]eat�XctukpV �Ct�tj� k;h�ea\O^�Vat�X[tuk;V % � � Kopt��;\,b�\zt�XcZa\,`

f(ν1) < f(τ)kp`

f(ν2) < f(τ)� ���TtuV]eawa|zXctukpVTkpVkpVa\{k;h=X[Zat�b²XW��ky|,\,~�~ub²hskp`���Zat�|}Z$XcZa\ft�Va\,½>w�gi~ut�XW�

tub�X[`dwa\���\Kkp�xX}g;t�VTXcZa\K`c\zbcwa~�X���º³\,`d\{|zkpqn\,b²X[Za\�XcZa\,kp`d\,qnb�k;h*e]tubc|z`c\OX[\f�$kp`dbc\�X[Za\zkp`d�;�

� �c�l�kc�� n@p�� n��M«O¬M�O« ,]� .�§j¬ «$ª[«�40*�/7,;ª & - ©}®('�)+/u«Sxx«�,0.8¦

f,"¦;§u¨�©Oª[«O¬ «

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a < b,;ª}«{ª[«5,0/q.z*�';�O«Oª[¨Q¨w*�©}¯¬s¯�,;¬

[a, b]©}®(.�¬-,;§j.]¨ .�®

©OªO§j¬C§G©5,0/qy(,0/ *�«}¨(®�{f � ¬s¯�«w. M(b) ↘M(a)

�� klql ��¥:tu`cb�Xµk;h�g;~u~.� ��\ ��t�~u~�e]\,qnkpVabdXc`[giXc\#X[Z�g�Xµ��\|�g;V"m8\,`�X[wa`d�

fXck¼k;�]X}g;t�V

g1 − 1

hswaV]|zX[t�kpV¼kpVK�#U h

τ (p+1) > σ(p) b[g�X[tub�^�\,b f(τ) ≤ f(σ)XcZa\,V¼��\ym�\z`R6

X[wa`d�f���Y`c\zma~�g;|ztuVao

f(τ)�>�

f(τ) − ε�Mkp`

f(σ)���

f(σ) + εhsk;`

ε ≥ 0bdq�gi~u~

\,Vakpw]opZ�� ��t�XcZakpw]Xf|}Z�g;V]optuVao$��Zatu|}Z!|z\,~u~�b(g;`d\�|z`ct�Xctu|,g;~G�UWV¡X[Z]\#b[g;qn\y��g��Àt�hσ(p)b[giXctubd^a\,b

f(τ (p+1)) 6= f(σ) 6= f(ν(p−1))hskp`K\,g;|}Z

τ (p+1) > σ > ν(p−1) �:X[Z]\,VF��\q�g��#m8\,`�X[wa`d�f���T|}Z�g;Vao;tuVao

f(σ)���

f(σ) ± ε�]hskp`

ε ≤ 0bdq�g;~�~�\zVakpwaopZ��a��t�X[Z�6

kpw]X�|}Z�g;Vaopt�VaoK��Zatu|}Z#|z\,~�~ublg;`c\³|,`dt�Xctu|�gi~G���:\,`�X[wa`d�atuV]ofX[Zat�bl��g��p�>��\Qe]kpV � X�|}Z�giVaop\

Va\,t�X[Za\z`M(a)

�8V]kp`M(b)

�l´²ZDw]bf : K → R

tub1 − 1

�U hf−1([a, b]) = ∅

´²Za\,VM(a) = M(b)

g;Vae�XcZa\,`d\7tub³Vak;XcZatuV]oyX[kyma`ck��;\;� ' XcZa\,`�6��tubc\i�]���#mag;`dXct�X[t�kpVat�Vao

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X[Za\z`c\Ktub�gµbct�Vaop~�\�VakpVa|z`ct�Xctu|,g;~�|z\,~u~σ��t�X[Z

f(σ) ∈ [a, b].

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�8����� ��� �q�������u� ������ ���������$ Â$#�Ct �

∃τ (p+1) > σ��t�X[Z

f(τ) ≤ f(σ)�

�stut �∃ν(p−1) < σ

��t�X[Zf(ν) ≥ f(σ)

�UWV£|�gibc\ �Ct �O� ��\$q(w]bdX_Z�g��p\

f(τ) < a�jhskp`

τg;Vae

σg;`d\TXW��k¼VakpVa|z`ct�Xctu|,g;~�|z\,~u~�b

t�V[a, b]�±��´²Z>wab

τ ⊆ M(a)�Yvxt�Va|,\

σtubfg$hCg;|z\#k;h

τ��\#Zag��;\

σ ⊆ M(a)g;V]e

|zkpVabd\,½>wa\,V�Xc~��M(a) = M(b).� o�g;t�V��]X[Z]\,`c\Kt�b²Vak;X[Z]tuVao(Xckyma`dk��p\;�

vxwamam8kpbd\E�Ctut ��t�b�X[`dwa\;� ��\,qnq�g % � � # X[\,~�~ubBwabQX[Z�g�X∀τ (p+1) > σ

�f(τ) > f(σ)

�UWVTm�gi`dX[t�|,wa~ug;`,�

f(τ) > b�Ck;XcZa\,`���tubd\

σgiVae

τg;`c\�t�V

[a, b]��Z]tu|}ZT|,kpV�Xc`[g;eat�|zXcb�X[Za\

Z��Dm8k;X[Za\zbct�b3�O� ���T~�\,qnq�g % � �D �]��\{Zag��;\∀τ > σ

�f(τ) > b

�¸´²Z>wabσ ∩M(a) = ∅.vDtuVa|z\;�Ctut �lt�b�X[`dwa\;�aXcZa\,`d\�\O�xtub�X[bν(p−1) < σ

��t�XcZf(ν) > f(σ)

��giVaeTtuV m�g;`�X[t�|,wa~ug;`f(ν) > b

�TU hν(p−1) 6= ν

t�bfgiVak;X[Z]\,`(p − 1)

6 hCgi|,\nk;hσ�=X[Za\zV

f(ν) < f(σ)�s|zh

e]\z^�Vat�X[t�kpV % � � C�Ctut �5�±� ´²Za\,`d\zhskp`d\f(ν) < a.´²Z>wab

νg;Vae$g;~u~�t�Xcb�hCg;|,\zbBgi`c\K|,kpV�X[g;tuVa\ze$t�V

M(a)�

��\zXσ(p) 6= σ

�8\²g;VakiX[Za\z`p6 hCg;|z\�k;h

M��t�XcZ

σ > ν�A¶�kpV]eat�XctukpVE�stj�=kih�e]\z^�Vat�X[t�kpV

% � � (t�qyma~�tu\zbf(σ) > f(ν) > b.

�l� ~u\zqyqyg % � �x �at�hσt�b�g;V��T|,\,~�~�bcw]|}Z$X[ZagiX

σ > ν��X[Z>wab

f(σ) > b��vxk

ν ∩M(a) = ∅.U X�hskp~�~uk���b�XcZ�giXM(b)

|,g;V$�8\�\O�xma`d\,bdbc\,eYgib�gneat�bs»dkpt�V�X²waVat�kpVM(b) = M(a) ∪ σ ∪ ν,��Z]\,`c\

νtub�g(hs`c\z\KhCgi|,\Kk;h

σ�¸¥*t�V�g;~u~��

M(b) ↘M(a)�

� �c l�kc�� nqp�� ��M«O¬M�O« ,�� .�§j¬W«ª[«�40*8/7,;ª & - ©}®('�)+/u«Sxx« ,0.8¦

f,!¦;§u¨�©Oª}«O¬W«

­¼®�ª[¨�« { *8.�©O¬C§G®(.F®(.M� � * )e)�®�¨�«

σ(p) §u¨L,©OªO§j¬C§G©5,0/:©}«w/j/*®�{�§j.8¦D«Sx p �A§j¬j¯f(σ) ∈ [a, b]

,0.8¦f−1([a, b])

©}®(.�¬-,;§j.]¨�.�®¼®�¬s¯�«Oª©OªO§j¬C§G©5,0/@)�®�§j.�¬s¨�� 1�¯�«w.M(b)

«}¨O¬�¯�®('�®�¬W®3)8°«���*D§jy(,0/u«w.�¬¸¬ ®

M(a)⋃

ep

ep,

�*¯�«Oª[«ep¦D«w.�®�¬W«}¨G,

p v ¦;§j'�«w.]¨O§G®(.+,0/M©}«w/j/ �A§j¬j¯��O®(*�.8¦e,;ªO° ep�

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� % ��������� ������������������������������ ���������!�#"%$

� klql � � b²t�V ma`ckDk;hMk;hMX[Z]\,kp`d\,q % � � � ��\Kq�g���g;bcbdwaqn\ftub

1− 1�A´²Z>wab²��\�|�g;V

^�Vaea′g;V]e

b′��t�XcZ

a ≤ a′ < b′ ≤ b��t�X[Z

σ = f−1([a′, b′]).

���$XcZa\,k;`c\,q % � � �M(b) ↘ M(b′)

g;V]eM(a′) ↘ M(a)

��bck t�XQbcw �y|z\,b�X[k�ma`ck��p\X[Z�giX

M(b′)Zakpqnk;X[k;m>��\,½>wat��;gi~u\,V�X²Xck

M(a′)⋃

ep

ep.

vxtuVa|z\σtubfg$|z`ct�Xctu|,g;~ |,\z~u~.�:t�h

τ (p+1) > σXcZa\,V

f(τ) > f(σ)g;Vae¡bck

f(τ) > b′�

�M\zqyqyg % � �x Xc\,~�~ub²wab�X[Z�g�X�t�hτtub³gy|z\,~�~�k;h

Mbcwa|}Z�XcZ�giX

τ > σX[Z]\,V

f(τ) > b′�

´²Za\,`d\zhskp`d\σ ∩M(a′) = ∅.

vxtuVa|z\σtub=g�|,`dt�X[t�|�g;~;|,\,~�~�k;h

M�iX[Z]\,Vfhsk;`:giV>�K|z\,~�~

ν(p−1) < σ��\�Z�g��;\

f(ν) < f(σ)g;VaeTbckf(ν) < a′.

¶�kpVabc\z½>wa\,V�X[~��ν ⊆M(a′)

g;Vaeσ ⊆M(a′)

�A´²Z]\,`c\Ohskp`c\

M(b′) = M(a′)⋃

σ

σ.

vxtuVa|z\σt�b²Zakpqn\,kpqnkp`dmaZatu|BX[k

ep�]X[Z]\{m]`ckDk;h*t�b²|,kpqnma~�\zX[\i�

���\OXmp(f)

�skp`7bdtuqnma~��mp

t�h�t�X7��t�~u~¸Vak;X7|�g;w]bc\#|zkpV]hswabdtukpV �K��\nX[Za\�VDw]q(��\z`k;h�|z`ct�X[tu|,g;~ |z\,~u~�b�k;h�t�Vaea\±�

p�n´²Za\

mp

� bfg;`c\_|�gi~u~u\ze X[Za\ ­F®�ª[¨�« .z*�';�O«Oª#®�{f� �

|,kpVabd\,½>wa\zVa|,\fk;h:XcZa\,k;`c\,qnb % � � � g;Vae % � � � tub� l�kl��"�#$&%�nqp�� �=«O¬

M�O«E,;� .�§j¬ «#ª[«�40*8/7,;ª�& - ©}®('�)+/u«Sxx«�,0.8¦

f,À¦;§u¨�©Oª[«O¬ «

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M§u¨n¯�®('�®�¬ ®3)8°F«���*D§jy(,0/u«w.�¬Q¬W®�, & - ©}®('�)+/u«Sx

�A§j¬s¯�«Sx ,D©O¬s/ °mp(f)

©}«w/j/ ¨(®�{{¦;§j'�«w.]¨O§G®(.p�

O�P���P�� �I^ ��a8_ U ��_�� ��\ Z�U !�U�_@a¤³\Ohs\,`c\zVa|,\zb�hskp`µX[Z]tubnbc\,|OX[t�kpVabyg;`c\/� ��� �sm�� ��� XckFm�� �] �(hskp`µX[Z]\$X[Z]\,kp`d\,q g;Vae � � ��C|}Z�g;m�� UR��hskp`²`d\,bcw]~�X[b�kpVTZak;qykp~�kpo;�;��¡\�^�`dbdX³Va\,\ze$bdkpqn\{e]\z^�Vat�X[t�kpVab�g;VaeTma`d\,~�tuqntuV�g;`��n`c\,bdwa~�Xcb,�bCcedgf�hjikhml+fonqp�� ���M«O¬

S�O« ,�{ *�.�©O¬.§G®(. {±ª[®(' ©}«Oª±¬-,;§j.]¨�)�,;§jª[¨ ®�{ ¨�)�,D©}«}¨¬W®¡¬j¯�«

§j.�¬ «�4D«Oª[¨��S§u¨ybdwa��g;eae]t�X[t��p\ | ª[«}¨�)�«}©O¬.§jy;«w/ °�g;eaeat�X[t��;\w}�§ {

∀ X ⊃ Y ⊃ Z��«f¯�,0y;«

S(X,Z) ≤ S(X, Y ) + S(Y, Z)| ª}«}¨�)�«}©O¬C§jy;«w/ °

=}0�

Page 21: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8����� ��� �q�������u� ������ ���������$ �Dº³\,`c\fg;`d\QXW��kntuqnm�k;`dX}giV>X�\O�]g;qnma~u\zb �

�gx ,0'�)+/u« � �bλ(X, Y ) � λ

XcZ��Q«O¬.¬C§ .z*�';�O«Oª7k;h(X, Y ) �

`[g;V]��k��p\z`Bg7^a\,~ueFk;h

Hλ(X, Y )�ahskp`�giV>�#m�g;tu`

(X, Y )bcwa|}ZX[Z�g�X³XcZatub²`cg;Va�#t�b�^�Vat�X[\;�

bλt�bKbcwa��gieaeat�Xct��;\n���¼\O�]g;qntuV]tuVao#XcZa\nhsk;~u~uk���t�Vao m�k;`dX[t�kpV k;h�X[Za\n\O�]g;|OX7bd\,½>wa\,V]|,\

hsk;`�X[Za\�X[`dtuma~�\X ⊃ Y ⊃ Z

. . .→ Hλ(Y, Z) → Hλ(X,Z) → Hλ(X, Y ) → . . .�jhskp`�gµma`ckDk;h:bc\z\{XcZa\,kp`d\,q Â$% � � m�� �&� ea\ � ��� �±��gx ,0'�)+/u« � �BU hAXcZa\7t�V>Xc\,op\z`cb

(bλ(X, Y ))g;`d\7g;~�~M�,\,`dk#�aw]XQgn^�V]t�X[\fV>waq(�8\,`,�3X[Z]\,V

��\{ea\z^�V]\KXcZa\ � *8/u«Oª(©}¯�,;ªS,D©O¬ «Oª±§u¨O¬C§G©����

χ(X, Y ) =

+∞∑

λ=0

(−1)λbλ(X, Y ).

´²Z]\�¹Awa~�\,`K|}Z�g;`cg;|zXc\,`ct�bdXctu|_tub{g;eae]t�X[t��p\i� � bfgTqygiXcXc\,`Kk;hlhCg;|zX,�:t�h�XW��k�giX{~�\�g;b�XKk;hhsk;~u~uk���t�Vaoµ¹Awa~u\z`²|}Z�g;`cg;|zXc\,`ct�bdXctu|Kg;`d\{e]\z^�Va\ze���X[Za\zVXcZa\�X[Zat�`ceTtubz��g;Vae

χ(X) = χ(Y ) + χ(X, Y ).´²Z]tubl|,kpqn\,blhs`ckpq�XcZa\�\±�agi|zX[V]\,bcb�g��xtukpq �s|zh � �]t�kpq � m�� Âp � � � �lg;VaeyXcZa\BhCgi|zXlX[Z�giXXcZa\K\O�]g;|OXBbd\,½>wa\zVa|,\(t�b�^�Vat�X[\;�� c���� nqp�� ���M«O¬

S�O«Q¨w* �3,�¦�¦;§j¬C§jy;«G,0.8¦

X0 ⊂ X1 ⊂ . . . ⊂ Xn

��1�¯�«w.

S(Xn, X0) ≤n∑

i=1

S(Xi, Xi−1).

j{S§u¨�,�¦�¦;§j¬.§jy;«�¬s¯�«w.�¬j¯�«µ«���*�,0/ §j¬.°µ¯�®(/�¦�¨��

� klql ���F\_ma`ck��p\_X[Z]tub����YtuVae]wa|zXctukpV kpVn�n¥�kp`

n = 1����\nZag��;\nXcZa\_\,½>w�g;~�t�XW�

giVae$t�hn = 2

�at�X � b²kpVa~��yX[Za\Kea\O^�Vat�X[tuk;V k;h*bdwa��g;e]eat�Xct��Dt�XW�;�U hMXcZa\Q`c\zbcwa~�X�t�b�X[`dwa\�hskp`

n− 1�xXcZa\,V

S(Xn−1, X0) ≤∑n−1

i=1 S(Xi, Xi−1)�A´²Za\z`c\ 6

hsk;`c\;�

S(Xn, X0) ≤ S(Xn−1, X0) + S(Xn, Xn−1) ≤n∑

i=1

S(Xi, Xi−1)

giVaeXcZa\K`c\zbcwa~�X�tub�X[`dwa\�hskp`n�

�F\Kma`ck��;\{XcZa\K`c\,bdwa~�X²XcZa\Kb[g;qn\Q��g�� hsk;`Sg;eaeat�X[t��p\;�

���\zXS(X, ∅) = S(X)

�A´:g;�>tuV]oX0 = ∅

t�VT~u\,qnqyg % � � �(��\Kop\OX

S(Xn) ≤n∑

i=1

S(Xi, Xi−1)� � � � � � �

��t�X[ZT\,½>w�gi~ut�XW��t�hSt�b�g;eaeat�X[t��;\;�

Page 22: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

� � ��������� ������������������������������ ���������!�#"%$

��\zXM�8\�gK^�Vat�Xc\B`d\,opw]~�g;`�¶�� |,k;qyma~�\O�yg;V]e

fgfeat�bc|,`d\zXc\��$kp`dbc\³hswaVa|zXctukpVykpV

K��t�XcZt�bckp~ugiX[\ze#|,`dt�Xctu|�gi~�m8kptuV�X[bz� ��\zX

a1 < . . . < ak

�8\�bcw]|}ZXcZ�giXMai

|,k;V>X[g;tuV]b\O�]g;|zXc~��

i|z`ct�X[tu|,g;~�|z\,~u~�b,�]g;Vae

Mak = M� ���#X[Z]\,kp`d\,q % � � � ��\fZ�g��;\

H∗(Mai ,Mai−1) = H∗(M

ai−1 ∪ eλi,Mai−1)= H∗(e

λi , eλi)

=

{ |,kD\ �n|ztu\,V�X³op`dkpwamTtuVTeat�qy\zVabct�kpVλi

t.� \i�F

% k;X[Z]\,`d��t�bc\��Za\,`d\

λi

tub¸X[Z]\�tuV]ea\O�yk;h�XcZa\�|z`ct�X[tu|,g;~�|,\z~u~g�shskp`lg7m]`ckDk;h�bd\,\�XcZa\,kp`d\,q  �]� � m�� � � � � �g;VaeT\O�x|,t�bct�kpV�g0�]t�kpq hskp`�gnZ]kpqyk;~ukpo;���±�

� mama~��Dt�Vao�� % � � � � ��X[k∅ = Ma0 ⊂ . . .Mak = M

��t�X[ZS = bλ

��\{Z�g��p\

bλ(M) ≤k∑

i=1

bλ(Mai ,Mai−1) = mλ;

��Za\,`d\mλ

tub�X[Za\l�$k;`cbc\¸V>waq(�8\,`�kihxt�Vaea\±�λ�Cg;b�g²q�g�XcX[\z`�k;h>hCg;|OX

dimFHλi(eλi , eλi) =

1�O�� mama~��DtuVao(XcZatub�hskp`dq(wa~ug7X[kµX[Za\K|,g;bc\

S = χ��\fkp�xX}g;t�V

χ(M) =

k∑

i=1

χ(Mai ,Mai−1) = m0 −m1 +m2 − . . .±mdim M

´²ZDw]b³��\KZ�g��p\{ma`ck��p\zV �� �c�l�kc�� n@p�� � ���Yc) ����l� �=c ��f�c��,�� � hjikhmc���� �=«O¬

M�O« ,�� .�§j¬W«¸ª[«�40*8/7,;ª�& -

©}®('�)+/u«Sx�,0.8¦f,¼¦;§u¨�©Oª[«O¬ «_­¼®�ª[¨�«�{ *8.�©O¬C§G®(.¢®(.

M�! j{

¦D«w.�®�¬W«}¨�¬j¯�«n­F®�ª[¨�«.z*8';�O«Oª7®�{K§j.8¦D«Sx

λ®�{f,0.8¦

bλ¬j¯�«

λ¬s¯ �Q«O¬C¬C§ .z*8';�O«Oª � ¬j¯�«w.

bλ(M) ≤ mλ

� � � � � � �+∞∑

λ=0

(−1)λbλ(M) =+∞∑

λ=0

(−1)λmλ

� � � � � � �

vD~utuo;Z>Xc~��ybcZ�gi`cm8\,`�tuVa\z½>w�g;~ut�X[t�\,b�|,g;V��\Kma`dk0�;\,V$�>��X[Za\�hskp~�~uk���tuVao(g;`copw]qy\zV>X,��!c���� nqp�� " 1�¯�« { *8.�©O¬C§G®(.

§u¨Q¨w*��3,�¦�¦;§j¬.§jy;« �*¯�«Oª}«

Sλ(X, Y ) = bλ(X, Y ) − bλ−1(X, Y ) + bλ−2(X, Y ) − . . .± b0(X, Y )

Page 23: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8����� ��� �q�������u� ������ ���������$ � �

� klql ����t��;\,V$g;VT\O�]g;|OXBbd\,½>wa\zVa|,\(kih:�;\,|zXckp`�bcm�gi|,\h

−→ Ai

−→ Bj

−→ Ck

−→ . . . −→ D −→ 0´²Z]\,V$XcZa\K`[g;V]��k;hhma~�wab²X[Za\K`[giVa��k;h

itub²\z½Dwag;~�X[kµX[Za\K`cg;Va�#k;h

A�A´²Za\,`d\zhskp`d\;�

rank(h) = rank(A) − rank(i)= rank(A) − rank(B) + rank(j)= rank(A) − rank(B) + rank(C) − rank(k)

. . .= rank(A) − rank(B) + rank(C) − . . .± rank(D).º³\,Va|z\BXcZa\�~ug;bdXl\O�xma`d\,bdbctuk;V#tub

≥ 0� $Bk��¾|,kpV]bctue]\,`¸X[Za\³Zakpqnkp~uk;o;�(\±�agi|zX�bd\,½>wa\zVa|,\

kih g(X[`dtuma~�\X ⊃ Y ⊃ Z

�¸´²Za\,VT��t�XcZHλ+1(X, Y )

∂−→ Hλ(Y, Z)��\{kp�]X}gituV

rank(∂) = bλ(Y, Z) − bλ(X,Z) + bλ(X, Y ) − bλ−1(Y, Z) + . . . ≥ 0.¶�k;~u~u\z|zXctuVaonX[\,`dqybz�Sλ(Y, Z) − Sλ(X,Z) + Sλ(X, Y ) ≥ 0

����Zatu|}Z |zkpqnma~u\OX[\fXcZa\m]`ckDk;h �

�� mama~��Dt�VaoµX[Zat�b²bcwa��gieaeat�Xct��;\Khsw]Va|zXctukpV X[kµXcZa\Kbcm�g;|z\,b∅ ⊂Ma1 ⊂ . . . ⊂ Mak

��\{kp�]X}gituV �� �c l�kc�� nqp�� % ��� i&kl+f+� ��l� ��c ��f�c��,�+$� hjikhmc�� � �M«O¬

M�O« ,�� .�§j¬ « ª}«�40*�/7,;ª

& - ©}®('�)+/u«Sx ,0.8¦f,£¦;§u¨�©Oª[«O¬ «$­F®�ª[¨�«�{ *8.�©O¬C§G®(.·®(.

M� m{

¦D«w.�®�¬W«}¨�¬j¯�«­¼®�ª[¨�«L.z*8';�O«Oª7®�{K§j.8¦D«Sx

λ®�{f,0.8¦

bλ¬s¯�«

λ¬s¯ ��«O¬.¬C§ .z*8';�O«Oª � ¬j¯�«w.

∀ λ ≥ 0, bλ(M) − bλ−1(M) + . . .± b0(M) ≤ mλ −mλ−1 + . . .±m0.� � � � � � ���«w';,;ª �0�l´²Z]\,bc\{tuV]\,½>w�g;~�t�X[t�\,b�g;`c\KbcZag;`cm8\,`²X[Zag;VXcZa\�ma`d\z�Dtukpw]b³k;Va\,bz�UWVThCg;|OX���g;e]eatuVao�� % � � � � ��hsk;`

λg;Vae

λ− 1��\{k;�]X}g;t�V � % � � � � �O�

¶�k;qym�gi`ctuV]oE� % � � � � ��hskp`λg;V]e

λ− 1��Z]\,`c\

λ > dim(M)��\{kp�]X}gituV�� % � � � � �±�

OVP���P�� ��`�\!WYXZ�_@a ^(� !�U�a�[ ��_ !�_��I^ ��a8_ � � ��[!8UR^(��a¹ �;\,`d�$¶�� |,kpqnma~�\O�

MZ�g;b�gµeat�bc|z`c\zXc\f�$kp`cbd\�hswaVa|zXctukpV��¸¥3k;`²\O�]g;qnma~u\i�aea\z^aVa\{g

�kp`cbd\Khsw]Va|zXctukpVf�>�#bc\zXdX[t�Vaoa�]hsk;`²\�g;|}Z

σ ∈ K�

f(σ) = dim(σ).�F\µbc\z\nXcZ�giX�\O�p\z`d�Y|,\z~u~ t�b�|,`dt�X[t�|�g;~.�f¶�kp`ckp~�~�g;`�� % � � � t�V�X[Za\_|,g;bc\_k;hl`c\zopwa~ug;`K¶��|zkpqnma~u\±�]\zbQX[\z~u~ub�wab�X[Z�giX�g$¶�� |,kpqnma~�\O�$��t�X[Z

mp

hCg;|z\,bKk;h�eatuqn\,VabdtukpVptubQZak(6

qnk;Xckpm��#\,½>wat��ig;~u\zV>X²Xck�gn¶�� |,kpqnma~u\±�#��t�X[Zmp

|z\,~u~�b²k;h:eatuqn\,VabdtukpVp�

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� � ��������� ������������������������������ ���������!�#"%$

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M�O«G,�& - ©}®('�)+/u«Sx],0.8¦

N ⊆M�O«G,_¨w*��O©}®('�)+/u«Sx � 1�¯�«w.

,0.�° ¦;§u¨�©Oª[«O¬ «�­¼®�ª[¨�«!{ *8.�©O¬C§G®(.¡®(.Mª}«}¨O¬Cª±§G©O¬s¨f¬W®C, ¦;§u¨�©Oª[«O¬W«K­F®�ª[¨�« { *�.�©O¬.§G®(.À®(.

N� j{

σ ⊆ N§u¨G, ©OªO§j¬C§G©5,0/:©}«w/j/0{�®�ª

f| ¬s¯�«7®�ªO§74;§j.+,0/({ *8.�©O¬C§G®(.0} � ¬j¯�«w.�§j¬¸§u¨(©Oª±§j¬.§G©5,0/{�®�ª�¬s¯�«{ª[«}¨O¬CªO§G©O¬C§G®(.+�

� klql �:´²Zatub²hskp~�~uk���b�eat�`c\,|OX[~���hs`ck;q ea\O^�Vat�XctukpV]b % � � _giVae % � � � ��º³\,`d\{t�b�gn~�\,qnq�g{��Zatu|}Zopt��;\,b�wab³gµ|,kpV��;\,`cbd\7k;h=X[Z]\{m]`c\z�Dt�kpwab�~u\zqyqyg]�

�!c���� nqp��@r �=«O¬M�O«�, | � .�§j¬W«�} & -r©}®('�)+/u«SxG,0.8¦

N ⊆M�O«�,Q¨w* �O©}®('�)+/u«Sx �

1�¯�«w. ,0.�°n¦;§u¨�©Oª[«O¬W«�­¼®�ª}¨�«g{ *8.�©O¬C§G®(.¼®(.N©5,0. �O«f«Sxp¬ «w.8¦D«c¦µ¬ ®;,�¦;§u¨�©Oª[«O¬ «B­F®�ª[¨�«

{ *8.�©O¬C§G®(.£®(.M§s�C«��T§ {

f§u¨;,�¦;§u¨�©Oª[«O¬ «7­F®�ª[¨�«V{ *�.�©O¬C§G®(. ®(.

N � ¬j¯�«w.¡¬j¯�«Oª[«_§u¨�,¦;§u¨�©Oª}«O¬W«Q­¼®�ª[¨�« { *8.�©O¬C§G®(.g®(.

M¨w*�©}¯¬j¯�,;¬

∀σ ⊆ N, g(σ) = f(σ).

� klql � �M\OXc = maxσ⊆Nf(σ)

� (�\O^�Va\gkpV

M���Tbc\zXdX[t�Vaoa�

g(σ) =

{f(σ) si σ ⊆ Nc + dim(σ) si σ * N

' V]\B\,g;bct�~��µ|}Za\z|}�yXcZ�giXgtub�g{e]tubc|z`c\OX[\��kp`cbd\BhswaVa|OX[t�kpVyk;V

MXcZ�giX�\O�DX[\zVaeab

f��´²Z]\n�$kp`cbd\7hswaVa|OX[t�kpV�|,k;VabdXc`cwa|OX[\,eYt�V�X[Za\(g;��k��;\µma`ckDk;hAqyg����8\7�p\z`d��tuVabdw���6

|,tu\zV�X��KUWVYm�g;`dXctu|zwa~�g;`z��\O�p\z`d�Y|,\z~u~:k;hM − N

tub�g#|z`ct�Xctu|,g;~M|,\,~�~G�K´²Za\z`c\µqyg��$\O�xt�bdX\O�DX[\zVabct�kpVab�Xck

M��t�XcZ�q�giV>�hs\z��\,`�|,`ct�X[t�|�g;~M|,\z~u~�b,��UWV�XcZa\7hskp~�~uk���tuV]oy~�\,qnq�g]����\

��tu~u~�bc\z\ft�X²tub²m8kpbcbdtu�a~�\Q��Za\,VM|zkp~u~ug;mabd\,b²kpV�X[k

N�

�!c���� nqp�� n��=«O¬M�O« , | � .�§j¬W«�}�& -�©}®('�)+/u«Sx�,0.8¦

N ⊆M�O«,f¨w* �O©}®('�)+/u«Sx

¨w*3©}¯¾¬j¯�,;¬M ↘ N

� �M«O¬f�O« ,¢¦;§u¨�©Oª[«O¬ «$­F®�ª[¨�«�{ *�.�©O¬C§G®(.�®(.

N,0.8¦

c =maxσ⊆Nf(σ)

�;1�¯�«w.f©5,0.��O«(«Sxp¬ «w.8¦D«c¦n¬ ®],n­¼®�ª[¨�« { *8.�©O¬C§G®(.F®(.

M�A§j¬j¯

N = M(c)

,0.8¦_¨w*3©}¯¬j¯�«Oª[«G,;ª[«L.�®#©Oª±§j¬.§G©5,0/*©}«w/j/ ¨�§j.M −N

� klql ��´²Za\Bm]`ckDk;h�t�bl���_t�Vaeawa|OX[tuk;Vnk;VnXcZa\BV>waq7��\z`�k;h�\,~�\,qn\,V�X}gi`d�_|,k;~u~�gimabc\zb¸g;Vaebck�t�X�bdw��n|,\zbQX[k�m]`ck��p\7XcZatub���Za\zV

M|,kp~�~�g;m]bc\,b³kpV�X[k

N����gybdtuVaop~�\f\z~u\zqy\zV>X[g;`d�

|,kp~�~�g;mabd\;�Bvxwam]m�kpbd\,bτtubBg�|,\z~u~Mk;h

M��t�X[Z�gyhs`c\z\fhCg;|z\

σ < τbdwa|}Z X[Z�g�X

Mt�b�g

eatubj»dkptuV�X²waV]tukpVM = N ∪ σ ∪ τ.

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�8����� ��� �q�������u� ������ ���������$ � �

(B\z^�Va\fgn�$kp`dbc\KhswaVa|zXctukpVgkpV

M���

g(ν) = f(ν), ν 6= σ, τg(τ) = c+ 1g(σ) = c+ 2

´²Z]\,Vgb[g�X[tub�^�\,b�X[Z]\{`d\,½>wat�`c\,ema`dkpm�\z`dXctu\,bz�

� l�kl��"�#$&% nqp� ��M«O¬∆n�O«�,

n v ¨O§j'�)+/u«Sx �A§j¬j¯§j¬j¨�¨O¬-,0.8¦e,;ªd¦�¬CªO§s,0.�40*�/7,;¬C§G®(. � ,0.8¦∆n

§j¬s¨��O®(*�.8¦e,;ª±° ��1�¯�«w.| §�}

∆n¯�,�¨G,n­¼®�ª[¨�« { *8.�©O¬C§G®(. �A§j¬j¯ «Sx ,D©O¬ / ° � ©Oª±§j¬.§G©5,0/�)�®�§j.�¬��

| §j§�}∆n¯�,�¨G,n­¼®�ª[¨�« { *8.�©O¬C§G®(. �A§j¬j¯ «Sx ,D©O¬ / ° � ©Oª±§j¬.§G©5,0/�)�®�§j.�¬j¨��

� klql ���stj��hskp~�~uk���b�hs`dkpq ~u\zqyqyg % � � � g;VaeTX[Z]\KhCgi|zX²X[ZagiX∆n|zkp~u~ug;mabd\,b²kpV�X[knkpVa\

kih*t�X[b��p\z`dXc\O����stut �=hskp~u~�k���b*hs`ckpq�~u\,qnqyg;b % � �  g;Vae % � � � �>g;Vaeyg;~�bck�X[Za\�hCg;|OXAX[Z�giX¸hskp`¸giV>�

(n−1)6

|z\,~�~σkih

∆n�∆n − σ

|,kp~�~�g;mabd\,b²kpV�XckykpV]\{kih*t�X[b��p\z`dXc\O����

OVP���P�� � ��_ !�U�a�[ ��_ !�_�� � \(!�U�_ � !�� _@[�!8^ � �_@Z ! \(� ! !���_�\ a�a8^�[ U��\"!8_ !���^�

�F\7��t�~u~Mea\z^aVa\µtuV�g�V�giXcwa`[g;~=��g��X[Za\7Vak;X[t�kpV�kih¸eatubd|,`c\OX[\(o;`[g;eat�\,V�X��;\,|OX[kp`Q^�\z~ueabXcZ�g;Va�>b³X[kyXcZa\{eatubd|,`c\OX[\(�$kp`cbd\{hsw]Va|zXctukpV���´:kyeakyXcZatubz�3��\f��t�~u~�ea`cg��·kpVa\ � b³tuVabdmat76`cgiX[t�kpV#hs`ck;q X[Z]\{e]\z^�Vat�X[t�kpVTk;h*o;`[g;eat�\,V�X²�p\z|zXckp`³^a\,~ueTkpVTbdqykDk;X[ZTqyg;Vat�hskp~ue����\zX

M�8\lg²op\,Va\z`[g;~D¶�� |,kpqnma~�\O�;�sVak;X=Va\z|,\,bdb[g;`��{`c\,o;wa~�g;` ��g;Vae

f�8\�g²eat�bc|z`c\zXc\

�kp`cbd\Khsw]Va|zXctukpVTkpVKX[Za\Kkpm8\,V|,\z~u~�b²k;h

M�

�¡\X[`��£Xck¡ea\O^�Va\ giV¢kp�D»d\z|zXVf : K → K

��Zat�|}Z¢tubyg¡eat�bc|,`d\zXc\�g;V�gi~ukpo¼k;h−∇f

X[Za\�op`cg;eat�\,V�X��;\,|zXckp`�^�\z~ue���giVaeTg;~ubdkΦf

XcZa\�g;bdbckD|,tugiX[\ze��k��K��¥�kp`�Vak��K�]��\gibcbcw]qy\

fgeat�bc|z`c\zXc\y�$kp`cbd\nhjwaVa|zXctukpV¼Z�g;bK�8\,\zVF^]�x\,e��*g;V]e¼��\n��`dt�Xc\

Vg;V]e

Φ�j��\Khskp`dop\zX²XcZa\Kbcw��µ�f�±�

´:k$opt��;\�gV�giXcwa`[g;~Ae]\z^�Vat�X[t�kpVFkih�X[Za\yeatubd|,`d\zX[\_�p\z|zXckp`f^�\z~ue��=��\n^�`cb�X(|zkpVabct�ea\,`XcZa\K�p\,`�X[t�|,\,bBk;h

M� �M\OX

v ∈ K0

��U hvtub³gy|z`ct�Xctu|,g;~�|,\z~u~.�a��\K��tubdZ�t�Xcb²�;gi~uwa\�X[ky��\

�z\,`dk�g;b�t�V bcqnkxk;XcZ$�$kp`cbd\�X[Za\zkp`d�;��vDkn��\Kbc\zXV (v) = 0.

U hvt�b�VakiXKg#|,`dt�X[t�|�g;~=|,\z~u~.��X[Z]\,V X[Za\z`c\µt�b�g#waVat�½Dw]\µ\,eaop\

e > v��t�X[Z

f(e) ≤f(v)

�³´²Za\f\,eaop\ebcm8\,|zt�^�\zb�X[Za\fwaVat�½Dw]\7eat�`c\,|OX[t�kpV�t�V$��Zat�|}Z

ftub³Vak;X�tuVa|z`c\,g;bct�Vao

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� � ��������� ������������������������������ ���������!�#"%$

¥*tuopw]`c\ � � � � � � (�t�`c\,|OX[t�kpVTk;h:XcZa\Kop`[gieatu\zV>X²�;\,|OX[kp`�^�\,~�e

�C|zhB¥*t�oa� % � � � � �O�¼v '−∇f(v)

bcZ]kpwa~ue!�8\e� �¡\�q(wabdX_V]k0��ma`d\,|ztubc\#XcZatubz� �¡\

��tubcZ�Xck#X[Zat�Va�k;hV (v)

g;bKg�eat�bc|z`c\zXc\(X}g;V]op\,V�XQ�;\,|zXckp`Q^�\z~ue m8kptuV�X[t�Vao#g���g���hs`ckpqv�A´²ZagiX³tub

V (v)tube��t�XcZ$g_|}Zakpbd\,V$kp`ct�\,V�X}giXctukpV��

´²Z>wab{��\ntuV�X[`dkxe]wa|,\_X[Za\_|}Z�gituV¡|zkpqnma~u\±� k;hM� ¥*t �Yg;V¼kp`ct�\,V�X}g�X[tuk;VYhsk;`{\,g;|}Z

|,\,~�~σ ∈ K

����\zXCi(M,Z)

�8\�X[Za\�hs`d\,\�g;�8\,~�t�g;V7op`ckpw]mµop\,Va\z`[giXc\,enk��p\,`Z�>�fX[Za\zbc\

kp`ct�\,V�X[\ze¡|,\z~u~�bfk;h�eatuqn\,V]bctuk;Vik;hM�T�F\yt�ea\,V�X[t�hj�

−σ��t�XcZ

σo;t��p\zVF��t�X[ZFXcZa\

kpmam8kpbct�X[\�kp`dtu\,V�X[giX[t�kpV�����\OX∂ea\,V]k;X[\KX[Za\Kwabdw�g;~��8kpwaVaeag;`d� k;m�\z`[giXckp`,�

∂ : Cp(m,Z) → Cp−1(M,Z).

´²Za\,V∂σ =

ν(p−1)<σ

ε(σ, ν)ν,

��Za\,`d\KXcZa\ε� b�g;`d\�tuV�X[\zop\,`db�|�g;~�~u\,e XcZa\KtuVa|ztuea\zVa|,\�V>waq(�8\,`db,�

U XKtubQ|zkpV��p\,V]tu\,V�XKX[kTt�V>Xc`ckDeawa|z\_g;V tuVaV]\,`�ma`dkxeaw]|zX<,>

kpVC∗

��� e]\,|,~ug;`ct�Vao�XcZa\|,\,~�~ub²k;h

MX[k_��\{g;VTkp`�X[Zak;Vakp`cqyg;~���g;bct�b,�¸´²Z>wab���\Kkp�]X[g;tuV

∂σ =∑

ν(p−1)<σ

< ∂σ, ν > ν.

´²Z�g;Va�>b_X[k�X[Z]tub,�¸��\|zkpqnma~u\OX[\ kpwa`µe]\z^�Vat�X[t�kpV!k;h³X[Za\Teat�bc|,`d\zXc\o;`[g;eat�\,V�Xµ�p\z|zXckp`^�\,~�e£k;V"XcZa\ �p\z`dX[t�|,\zb,�"U h

v ∈ K0

t�bµVak;Xng |,`dt�Xctu|�gi~�|,\z~u~.�¸g;Vaee ∈ K1

�p\z`ct�^a\,be > v

�f(e) ≤ f(v)

�]��\{bc\OXV (v) = ±e

����Za\z`c\�X[Z]\{bdtuopVTt�b²ea\zXc\,`cqntuV]\,ebckµX[Z�g�X< ∂(V (v)), v >= −1.

Vak�� ��\ea\O^�Va\TX[Za\ �Ceat�bc|z`c\zXc\Xctuqn\ � ak0�Φg;~ubdkYtuV gYVagiX[wa`cg;~���g��;�¢U h

vtubng

|,`ct�X[t�|�g;~�|,\,~�~G�DX[Za\zVV (v) = 0

g;Vaebckvtub�^]�x\,ewaVae]\,`�X[Za\Kop`cg;eatu\zV�X �k��K��tG� \;�

Φ(v) = v.

Page 27: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8����� ��� �q�������u� ������ ���������$ �

U hvt�b³V]k;X�|,`dt�X[t�|�g;~.�ag;V]e

V (v) = ±e��X[Za\zV

vbcZakpw]~ue �k��5X[kµXcZa\ � k;X[Z]\,`�\,Vae � k;h

e�¸´²Z�g�X�tub,�

Φ(v) = v + ∂(V (v)).$³k;X[\�X[Z�giX*X[Zat�b*hskp`cq7wa~�g³Zakp~ue]b*hskp` gi~u~>�p\,`�X[t�|,\,bz����\OX[Za\z`l|z`ct�X[tu|,g;~>kp` VakiX��A�kp`c\zk0�;\,`z�t�hMX[Za\K|,\z~u~�t�b²Vak;X�|,`dt�Xctu|�gi~G�>X[Za\zV�t�X�Z�g;b³g;|zXcw�g;~�~��yqyk��p\ze$bdtuVa|z\

< Φ(v), v >= 0�

�¡\�\±�DX[\,V]eVg;Vae

Φ~�tuVa\,g;`c~��yXcknqyg;mab²k;V|}Zag;tuVab

V : C0(M,Z) −→ C1(M,Z)

Φ : C0(M,Z) −→ C0(M,Z).�F\KVak��5\±�xXc\,Vae

VX[k_Zatuo;Za\,`²eat�qy\zVabct�kpV |,\z~u~�b,�

b�cedgf�hjikhml f nqp� �M«O¬σ�O«%,

p v ©}«w/j/=¦D« M | �A§j¬s¯ ,���xx«c¦®�ª±§G«w.�¬-,;¬C§G®(.0}0� j{{¬s¯�«Oª[«§u¨τ (p+1) > σ

�A§j¬j¯f(τ) ≤ f(σ) � ��«Q¨�«O¬

V (σ) = − < ∂τ, σ > τ.

| .�®�¬W«7¬s¯�,;¬σ'�*>¨O¬��O« ,Tª}«�40*�/7,;ª {=,D©}«n®�{

τ| {�®�ª

fª}«}¨�)�«}©O¬j¨f¬s¯�«�)�,;ª±¬.§s,0/A®�ªd¦D«Oªn®(.

§jª±ª[«�40*�/7,;ª {=,D©}«}¨7®�{τ}],0.8¦n¨�®

< ∂τ, σ >= ±1}0�

j{�¬j¯�«Oª}«{§u¨L.�®7¨w*3©}¯τ � ¬s¯�«w.���«Q¨�«O¬

V (σ) = 0.

�=®�ª7«5,D©}¯p � ��«(«Sxp¬ «w.8¦ V / §j.�«5,;ª / °n¬W®],]';,w)

V : Cp(M,Z) −→ Cp+1(M,Z).

��«w';,;ª �0� $³k;X[\QXcZ�giXV (σ) = 0

��t�X[Zak;w]Xσ^x�]\ze�waV]ea\,`²X[Z]\ ak0�K�

UWV�^aopwa`c\ % � � � � ��\(Zag��;\V (e1) = 0

� $B\z�;\,`�X[Za\z~u\,bdbe1

q7wabdX�Vak;XQ�8\7^]�x\ze waVaea\z`XcZa\ �k��

Φ�#´²Za\n��k;waVae�g;`��Yk;h

e1�Ct.� \i��t�Xcb�XW��k�p\,`�X[t�|,\,b � � qnk��p\neak���V���g;`deab � �

´²Z]tubTtu~�~uwab�X[`[g�X[\,bnX[Z�g�X X[Za\z`c\¼g;`c\¼gÀX}giVaop\,V�X$g;Vae gÀX[`cg;Vab��p\,`db[g;~K|zkpqym8kpVa\zV�XTk;hXcZa\(op`cg;eat�\,V�X

−∇f(v)�V (σ)

tub³X[Za\7Xc`[g;V]bd�p\z`cbcg;~A|,kpqnm8kpVa\,V�XQg;VaeV (∂σ)

tub³X[Za\X[g;Vaop\zV�XBk;Va\��CZa\z`c\���\{|�g�X[|}ZTX[Za\Kqnk0�;\,qn\,V�X³k;h=X[Za\��p\z`dXctu|,\zb3�±�b�cedgf�hjikhml f nqp� ���=®�ªG,0.�°®�ª±§G«w.�¬ «c¦�{=,D©}«

σ��«(¦D«m� .�«{¬s¯�« | ¦;§u¨�©Oª[«O¬W«{¬.§j'�«�}%4;ªS, v¦;§G«w.�¬���® �

Φ�,°

Φ(σ) = σ + ∂V (σ) + V (∂σ)®�ª � '�®�ª[«Q¨w*�©}©O§j.�©O¬ / ° �

Φ = 1 + ∂V + V ∂.

Page 28: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

��� ��������� ������������������������������ ���������!�#"%$

¥*tuopw]`c\ � � � � � �µ¶�\,~�~Ahskp`���Zatu|}Z¡XcZa\nop`[g;e]tu\,V�Xftub{V>wa~u~A��t�XcZakpw]X{�8\,t�Vao$^]�x\zeÀwaVaea\z`X[Za\ �k��

¥*tuo;wa`c\ � � � � � � ��`[g;e]tu\,V�X²�;\,|zXckp`²^�\,~�e$kih:XcZa\Kwamam8\,`�\,eao;\

¥:tuopwa`d\ � � � � � �¸¹ �xma`d\,bdbctuk;V$k;h=X[Z]\ ak0�Φ(e)

��x ,0'�)+/u«�¥*t�opwa`d\,b % � � � � g;Vae % � � � � tu~�~uwab�X[`cgiX[\{X[Za\µop`cg;eat�\,V�X��p\z|zX[k;`{^�\z~ue¼g;Vae XcZa\�k���hskp`�gµ|,\z~u~

ekp`ct�\,V�X[\zehs`dkpq XcZa\K~u\OhjX²X[kµX[Za\K`dtuopZ�X��

´²Z]\Qq�gituV_ma`ck;m�\z`dX[t�\,b�k;hVgiVae

Φg;`d\B|zkpV�X}g;t�Va\,e#t�VnXcZa\³hskp~u~�k0��t�Vao�X[Za\zkp`c\zqybz�

� �c�l�kc�� n@p�� � | � }V ◦ V = 0

Page 29: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8����� ��� �q�������u� ������ ���������$ � #

| � } m{σ§u¨�,0.¼®�ªO§G«w.�¬W«c¦

p v ©}«w/j/:®�{ M � ¬s¯�«w.#{ν(p−1)|V (ν) = ±σ} ≤ 1.

| � } m{σ§u¨�,0.¼®�ªO§G«w.�¬W«c¦

p v ©}«w/j/:®�{ M � ¬s¯�«w.σ§u¨(©Oª±§j¬C§G©5,0/

⇔ [σ /∈ Im(V ),0.8¦

V (σ) = 0].� klql �G�  ��U h

V (ν(p−1)) = ±σ(p) X[Za\zV ν < σgiVae

f(ν) ≥ f(σ)� ���¼~u\zqyqyg

% � � # �]X[Za\z`c\Ktub²V]kτ > σ

��t�X[Zf(τ) ≤ f(σ)

�A´²Z>wabV (σ) = 0

�� � ��U h

V (ν(p−1)) = ±σ(p) XcZa\,V ν < σg;Vae

f(ν) ≥ f(σ)� ����|zkpVaeat�X[tuk;V �Ct�tj�³kih

e]\z^�Vat�X[t�kpV % � � x�νtub�waVat�½>wa\;�

� � ��¥�`ckpq ea\z^aVat�XctukpV % � � � �σt�b²|,`ct�X[t�|�g;~�t�h*g;VaeTkpVa~��#t�h

�Ct �@ ν(p−1) < σ

�f(ν) ≥ f(σ)

giVae�stut �

@ τ (p+1) > σ�f(τ) ≤ f(σ)

�´²Z]\,bc\{|,k;Vaeat�XctukpV]b�g;`c\�\z½Dw]t��ig;~u\zV�X³Xck�Ct �

@ ν(p−1) � V (ν) = ±σg;Vae

�stut �@ τ (p+1) � V (σ) = ±τ

�t.� \i��Ct �

σ /∈ Im(V )g;V]e

�stut �V (σ) = 0

��

� �c l�kc�� nqp�� � | � }Φ∂ = ∂Φ

��M«O¬

σ1, . . . , σr

¦D«w.�®�¬W«K¬j¯�«p v ©}«w/j/ ¨(®�{ M �A§j¬s¯ ,T©}¯�®�¨�«w.¼®�ªO§G«w.�¬�,;¬C§G®(.+� -7ª±§j¬ «

Φ(σi) =∑

j

aijσj.

| � }∀ i � aii = 0

®�ª1 � ,0.8¦ aii = 1

§ {G,0.8¦T®(.z/ °n§ {σi

§u¨G,©Oª±§j¬C§G©5,0/:©}«w/j/7�| � } m{

i 6= j � ¬j¯�«w. aij ∈ Z� m{

i 6= j,0.8¦

aij 6= 0¬j¯�«w.

f(σj) < f(σi)�

� klql ���  �²vxt�Va|,\Φ = 1 + ∂V + V ∂

�a��\{Z�g��p\Φ∂ = (1 + ∂V + V ∂)∂ = ∂ + ∂V ∂ + V ∂2 = ∂ + ∂V ∂

∂Φ = ∂(1 + V ∂ + ∂V ) = ∂ + ∂V ∂ + ∂2V = ∂ + ∂V ∂.�F\{ma`ck��p\�� � �²g;V]e � � �²bdtuq(w]~�X}giVa\,kpw]bc~��;��^�`dbdX,�8bdtuVa|z\∂giVae

V��kiX[Zq�g;mtuV�Xc\,op\z`

|}Zag;tuVab²Xcknt�V�X[\,o;\,`²|}Z�g;t�Vab,��\,g;|}Zaij ∈ Z

��l�7X[Z]\,kp`d\,q % � �&�]�;\�g;|}Z_|,\z~u~

σpbcgiX[t�bd^�\zb¸\O�]g;|zXc~��7kpVa\²k;h3X[Z]\�hskp~u~�k0��t�Vao�m]`ckpm8\,`�X[tu\zb �

Page 30: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�&% ��������� ������������������������������ ���������!�#"%$

�stj�σtub�gµ|,`dt�X[t�|�g;~�|,\,~�~

�Ctut �±σ ∈ Im(V )

�Ct�tut �V (σ) 6= 0

��¡\K\O�]g;qntuV]\�\�g;|}Z|�g;bd\KtuVaea\zm�\zVaea\zV>Xc~�� Xcknm]`ck��p\KX[Za\�XcZa\,kp`d\,q��Ctj�lU h

σtub²|z`ct�Xctu|,g;~�XcZa\,V

V (σ) = 0��bdk

Φ(σ) = σ + V (∂σ) = σ +∑

ν(p−1)<σ

< ∂σ, ν > V (ν).

vxtuVa|z\σt�b |,`dt�X[t�|�g;~.�

∀ ν(p−1) < σ�f(ν) < f(σ)

� ¥�kp` \�gi|}Z bdwa|}Zν��\,t�XcZa\,`

V (ν) = 0kp`V (ν) = σ(p) ��t�X[Z

f(σ) ≤ f(ν) < f(σ).

´²ZDw]bΦ(σ) = σ +

∑aeσσ

��Za\,`d\aeσ 6= 0 ⇒ f(σ) < f(σ)

��Ctut ��vxw]mam�k;bc\

±σ ∈ Im(V ) ⊆ Ker(V )�Cg;Vaebck

V (σ) = 0�O�¸´²Za\,V

Φ(σ) = σ + V (∂σ) = σ +∑

ν(p−1)<σ

< ∂σ, ν > V (ν).

���yXcZa\,k;`c\,q % � �&��� � �O�xXcZa\,`d\�tub�\O�]g;|zXc~��ykpVa\(p− 1)

6W|z\,~u~ν < σ

��t�X[ZV (ν) = ±σ�shskp`

σ ∈ Im(V )�Qg;Vae

< ∂σ, ν > V (ν) = −σ�C|zhlea\O^�Vat�X[tuk;V % � � � �O�7�$kp`d\,k��p\z`,�

t�hν 6= ν < σ

X[Za\zVY\,t�XcZa\,`V (ν) = 0

kp`V (ν) = σ

��t�XcZf(σ) ≤ f(ν) < f(σ)

�´²ZDw]b

Φ(σ) =∑

eσ(p)

aeσσ,

k ���aeσ 6= 0 ⇒ f(σ) < f(σ)

��Ctut�tj�lvxwamam8kpbd\

V (σ) = − < ∂τ, σ > τ 6= 0�A´²Za\zV

Φ(σ) = σ + ∂(V (σ)) + V (∂σ).

vxtuVa|z\V (σ) 6= 0

�±σ /∈ Im(V )

� ´²Z>wab∀ ν(p−1) < σ

�²\,t�X[Za\z`V (ν) = 0

kp`V (ν) = ±σ

g��p\z|f(σ) ≤ f(ν) < f(σ)

�l�$k;`c\,k��;\,`

∂(V (σ)) = − < ∂τ, σ > ∂τ = − < ∂τ, σ >2 σ +∑

beσσ = −σ +∑

beσσ

k ���beσ 6= 0 ⇒ f(σ) < f(τ) ≤ f(σ)

�´²Zatub²|zkpqym]~u\zXc\QX[Za\Kma`dkxkih �

�' Va\{|�giV�b[g�� �C��� XcZa\,kp`d\,q % � � ��X[ZagiXΦea\,|z`c\,g;bc\zb

f�8g;Vae

σ ⊆ Φ(σ)t�h*g;Vae

kpVa~���t�hσtub²|,`dt�Xctu|�gi~G� ¥�kp`�\O�]g;qnma~u\�|zkpVabct�ea\,`³¥*t�oa� % � � � � X[Za\zV

Φ(e1) ⊇ e1

Page 31: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8����� ��� �q�������u� ������ ���������$ �xÂ

Φ(e2) ⊆ e2

Φ(e3) ⊆ e3�kp`c\Kma`d\,|,t�bc\z~��p�Φ(e1) = e1 + e2 + e3

Φ(e2) = Φ(e3) = 0.

Page 32: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�e� ��������� ������������������������������ ���������!�#"%$

9;:�� � � � H IL?*S � D�IGF KCJ ��� PfRCA LzRµN�P ?*LzPfRµJ D �TPKL,RTS�M\OX

M�8\TgY^aVat�Xc\¶�� |,k;qyma~�\O��� ��\zX

CΦp (M,Z)

e]\,Vak;Xc\ X[Za\Φ6 tuV��ig;`ctug;V�X

p6

|}Z�g;tuV]b³kihM�aXcZ�giX�t�b

CΦp (M,Z) = {c ∈ Cp(M,Z)|Φ(c) = c}.

¥3`dkpq�X[Za\zkp`c\zq % � � D�*X[Za\���kpw]Vae�g;`��¡kpm8\,`cgiX[k;`∂q�gimab

CΦp

X[kCΦ

p−1

��´²Z>wab(��\Z�g��p\fg_e]t���\,`c\zV�X[t�gi~�|,k;qyma~�\O�

CΦ : 0 −→ CΦn (M,Z)

∂−→ CΦ

n−1(M,Z)∂

−→ . . .∂

−→ CΦ0 (M,Z) −→ 0.

� � � � �  �´²Za\K|,kpqnma~u\±�

CΦ tub²|,g;~u~�\,e#XcZa\(­F®�ª[¨�«µ©}®('�)+/u«Sxp�´²Za\fo;k�g;~�k;h:X[Zat�b³bd\,|OX[tuk;V�tub²Xckyma`dk��p\fXcZ�giX�X[Z]\fZak;qykp~�kpo;�yk;hCΦ tub�ma`d\,|ztubc\z~�� XcZa\|,\,~�~�Zak;qykp~�kpo;�Yk;h

M�¼´²Za\#^�`dbdXµb�X[\zm£t�b7X[k t�V��p\,b�X[t�o�giX[\yX[Za\ b�X}g;�at�~ut���giXctukpVYqyg;m

Φ∞ : C∗ −→ C∗

opt��;\,V���Φ∞ = lim

N→∞ΦN

�$kp`c\�ma`c\z|,tubd\,~��p� ��\#��tu~u~lma`dk0�;\#X[Z�g�X(X[Za\z`c\#t�bN ∈ N

~�gi`cop\y\,Vakpw]opZ!bcwa|}Z"X[ZagiXΦ∞ = ΦN

�!c���� nqp�� "��=«O¬c ∈ CΦ

p (M,Z),0.8¦ �Aª±§j¬W«

c =∑

σ∈Kpaσσ� �M«O¬

σ∗ =,0.�° ';, xp§j'µ§���«Oª7®�{

{f(σ)|aσ 6= 0}.

1�¯�«w.σ∗ §u¨�,T©Oª±§j¬C§G©5,0/*©}«w/j/:®�{ f �

� klql � vxtuVa|z\ctub

Φ6 tuV��ig;`ctug;V�X��

c = Φ(c) =∑

σ∈KpaσΦ(σ)

�¸´²Za\z`c\Ohskp`c\i�

aσ∗ =< c, σ∗ >=∑

σ∈Kp

aσ < Φ(σ), σ∗ > .

hs`ckpq�X[Za\zkp`c\zq % � � C� � �±�at�hσ 6= σ∗ \zX f(σ) ≤ f(σ∗)

XcZa\,V

< Φ(σ), σ∗ >= 0.

´²ZDw]b0 6= aσ∗ = aσ∗ < Φ(σ∗), σ∗ >

�²vxk< Φ(σ∗), σ∗ >6= 0

��´²Z]\,kp`d\,q % � � �� � �tuqnma~ut�\,b�XcZ�giX

σ∗ tub�gµ|z`ct�Xctu|,g;~�|z\,~u~.��

� �c�l�kc�� n@p�� %∃N ∈ N

/7,;ª 4D«(«w.�®(* 4�¯ΦN = ΦN+1 = . . . = Φ∞ �

Page 33: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8���������� � ��� �������� � ����Y��� � ��������������� ��������� � � �

� klql �*´²Z]\fma`dkxkih:t�b²���TtuVaeaw]|zX[t�kpV kpVr = #{σ ∈ K|f(σ) < f(σ)}

�vDwamam8kpbc\

r = 0�pX[Za\zV����µX[Za\zkp`c\zq % � � x��\,t�X[Za\z`

Φ(σ) = σk;`

Φ(σ) = 0� UWVn\,t�XcZa\,`

|,g;bc\Φ(σ) = Φ∞(σ)

�vDwamam8kpbc\KX[Za\Kma`dkpm8\,`dXW�#Zakp~ue]b�hskp`

r − 1�l´²Z]\,VThskp`

r��\fZag��;\{XW��kn|�g;bd\,b �

�stj�σtub²V]k;X�|,`dt�X[t�|�g;~.�*´²Za\zV���� X[Z]\,kp`d\,q % � �

Φ(σ) =∑

f(eσ)<f(σ)

aeσσ.

�l�¡t�Vaeawa|OX[tuk;V��∃Neσ ∈ N

bdwa|}Z!X[Z�giXΦNeσ(σ)

t�bΦ6 tuV��ig;`ctug;V�Xf��Za\zVa\z�;\,`

f(σ) <f(σ)

�A´²Za\,`d\zhskp`d\ΦMax{Neσ}

tubΦ6 tuV��ig;`ctug;V�XL�

Max{Neσ}\O�xtub�X[b²hskp`

Mtub�^�Vat�X[\ �O�

�stut �σtub²|,`dt�Xctu|�gi~G����\OX

c = V (∂σ)�¸´²Za\zV

Φm(σ) = σ + c + Φ(c) + . . .+ Φm−1(c).U X{hskp~�~uk���bQX[Z�g�X

Φm(σ)t�b

Φ6WtuV��ig;`dt�g;V�XKt�h�g;VaeFkpVa~�� t�h

∃N ∈ N�ΦN (c) = 0

� �bd\,\zVyt�Vµma`ckDk;h8k;h3X[Za\zkp`c\zq % � � x�

ctub*XcZa\²bcwaq�k;h

p6W|z\,~�~ub

σbcw]|}ZnXcZ�giX

f(σ) < f(σ)�

�l� tuVae]wa|zXctukpV��∃N ∈ N

�Φ

eN(c)tub

Φ6 tuV��ig;`ctug;V�X��

�F\KVak��5kp�]bc\,`��p\KX[Z�g�Xc ∈ Im(V )

giVaeIm(V )

t�bΦ6WtuV��ig;`dt�g;V�X��]bctuV]|,\

ΦV = (1 + ∂V + V ∂)V = V (1 + ∂V )

�s|zh¸X[Z]\,kp`d\,q % � �&���  �5�±�K´²ZDw]bΦ

eN ∈ Im(V )� ����XcZa\,k;`c\,q % � �&� � � �O��XcZa\µtuqyg;op\

kihVtub�kp`�X[Zak;opkpV�g;~:X[k X[Za\n|,`dt�Xctu|�gi~:hCg;|z\,b{g;VaeYX[Za\z`c\Ohskp`c\

ΦeN(c)

t�bKgp6W|}Z�g;t�V

Φ6

t�V��;gi`ct�giV>XM��Zat�|}Z7t�b:kp`�X[Zak;opkpV�g;~pXckBXcZa\�|z`ct�Xctu|,g;~;hCg;|z\,b,� ���K~u\zqyqyg % � � � �Φ

eN (c) = 0�

��gx ,0'�)+/u«��B´²Z]\f^�opw]`c\ % � � �  t�~u~�wabdXc`[giXc\,b²X[Z�g�XNtub�^�Vat�Xc\{hsk;`�~�gi`cop\K\,V]kpwaopZ �sZa\,`d\

N �� hsk;`�X[Za\K|,\z~u~

e�±�

¥:tuopwa`d\ � � � �  �A¥�kp`�X[Za\K|z\,~u~e�N = 2

����X[Za\zkp`c\zq % � � # �]X[Z]\,`c\Kt�b�g;VN ∈ N

~�g;`dop\�\,V]kpwaopZTbck_X[ZagiX³\z�p\z`d�T|}Z�gituVc�

ΦN(c) = ΦN+1(c) = ΦN+2(c) = . . . .

Page 34: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

� � ��������� ������������������������������ ���������!�#"%$

�M\OXΦ∞(c)

ea\zVak;X[\²XcZatubΦ6 tuV��ig;`ctug;V�X |}Z�g;t�V�� ´²Za\zVnhsk;` \�g;|}Z

p ∈ N�p��\�Z�g��p\³qyg;mab

Φ∞ : Cp(M,Z) −→ CΦp (M,Z)

i : CΦp (M,Z) ↪→ Cp(M,Z)��Za\,`d\

itubMX[Z]\�V�giXcwa`[g;~DtuV]|,~uw]bctuk;V�� $Bk;Xc\�X[Z�g�X

Φ∞◦itubMX[Za\�t�ea\,V�Xct�XW�{kpV

CΦp (M,Z)

��¡\KZ�g��p\KVak��5XcZa\�hskp~u~�k���tuVao{X[Za\zkp`c\zq �

� �c�l�kc�� n@p ��� �M«O¬CΦ

¦D«w.�®�¬W«K¬s¯�«�­¼®�ª[¨�«µ©}®('�)+/u«Sx |�� � � � � }0�;1�¯�«w.∀p ∈ N

Hp(CΦ∗ ) ∼= Hp(M,Z).

� klql � ¶�kpVabct�ea\,`²XcZa\�hskp~u~�k0��t�Vao(|zkpqyq7w]X}giXct��;\Qeat�giop`[g;q �

0 −→ Cn(M,Z)∂

−→ Cn−1(M,Z)∂

−→ . . .∂

−→ C0(M,Z) −→ 0↓Φ∞↑i ↓Φ∞↑i ↓Φ∞↑i

0 −→ CΦn (M,Z)

∂−→ CΦ

n−1(M,Z)∂

−→ . . .∂

−→ CΦ0 (M,Z) −→ 0

�M\OXΦ∞

∗ : H∗(M,Z) −→ H∗(CΦ∗ )

i∗ : H∗(CΦ∗ ) −→ H∗(M,Z)ea\,VakiX[\KX[Za\{tuVae]wa|,\ze�qyg;mab�kpVZakpqnkp~ukpoi�p� ' wa`�opk�g;~�t�b�XckybdZak���X[Z�g�X

i∗g;V]e

Φ∞∗g;`c\³tubdkpqyk;`cmaZat�bcqnb,�=UWVyhCg;|OX��>��\³��tu~�~aX[Z�giXlX[Z]\z�yg;`c\³tuV��;\,`cbd\,b�k;h�\�gi|}Z#k;X[Z]\,`,�AvxtuVa|z\

Φ∞ ◦ i = 1��\{Zag��;\

1 = (Φ∞ ◦ i)∗ = Φ∞∗ ◦ i∗´:k_bc\z\{XcZ�giX

i∗ ◦ Φ∞∗ = 1

��t�X�tub²bcw �y|ztu\zV>X³X[kµ^�V]e�g;Vkpm8\,`cgiX[kp`L : C∗(M,Z) −→ C∗−1(M,Z)

bcwa|}ZX[Z�giX1 − i ◦ Φ∞ = ∂L + L∂.

� bMg²q�g�XcX[\z`�k;h>hCg;|OX��0|,~�kpbc\zeKhsk;`cqnb���k;wa~ueK�8\Aq�g;m]m�\zeKXck�\O�]g;|zX�hsk;`cqnb����1−i◦Φ∞ �g;VaeFXcZatubfq�gim¼��kpwa~�eÀ��\nX[Za\y�,\z`ck�qyg;mFkpVÀZakpqnkp~�kpo;�p� vxtuVa|z\

itubKX[Za\ytuea\zV>Xct�XW�

q�g;m k;V$|}Z�gitub³g;VaeΦ∞ = ΦN

hskp`²bckpqn\N~�gi`cop\�\zVakpwaopZ��

1 − i ◦ Φ∞ = 1 − ΦN = (1 − Φ)(1 + Φ + Φ2 + . . .+ ΦN−1)= (−∂V − V ∂)(1 + Φ + Φ2 + . . .+ ΦN−1)= ∂[−V (1 + Φ + Φ2 + . . .+ ΦN−1)] + [−V (1 + Φ + Φ2 + . . .+ ΦN−1)]∂

�s��\{w]bc\,eΦ∂ = ∂Φ

�±�L = [−V (1 + Φ + Φ2 + . . .+ ΦN−1)]

tub�gib³e]\,bct�`c\ze���

Page 35: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8���������� � ��� �������� � ����Y��� � ��������������� ��������� � �&�

OVP ��PSQ � ��_ �I^ ��a8_ [2^!WYXZ�_2` \(� ! [�� U !8UR[ \ Z�XG^!U � !8a��\OX

M��\Tg�^�V]t�X[\ ¶�� |zkpqnma~u\±���Y´²Za\Topkpg;~�kih³XcZatub7bc\,|OX[t�kpV!tub7Xck ma`ck��;\#X[Z�giX

XcZa\_�$k;`cbc\7|,kpqnma~u\±�|�g;V �8\7\±�xma`c\zbcbc\ze¼XcZ�g;Va�>bQXck#|,`dt�X[t�|�g;~�|z\,~�~ub,�³�¡\f^�`cb�X�ma`ck��;\XcZ�giX�XcZa\�bcmag;|,\Kk;h

Φ6Wt�V��;gi`ct�giV>X�|}Z�g;t�Vab�tub�|�g;Vak;Vatu|,g;~u~��yt�bckpqnkp`dmaZatu|�X[k(XcZa\�bcmag;VTk;h

XcZa\µ|,`dt�Xctu|�gi~:|z\,~u~�b,�K´²Za\zV�����\µ��tu~�~=^aVae¼g;V \O�xma~�tu|,t�XQ\±�]m]`c\,bdbct�kpVFkihlXcZa\µ�8kpwaVae�gi`d�k;m�\z`[giXckp`�k;h*XcZa\(�kp`cbd\f|zkpqym]~u\O�#X[Z�g;V]�Db�X[knX[Za\{|,`dt�Xctu|�gi~�|,\z~u~ub�giVae$XcZa\fV]k;X[t�kpVTk;ho;`[g;eat�\,V�X�m�giXcZ��¥�kp`�\�gi|}Z

p ∈ N��~u\OX

Mp

ea\zVak;X[\�XcZa\Kbcm�g;Vk;h=X[Za\K|z`ct�Xctu|,g;~p6W|,\z~u~�b,�at.� \i�

Mp = {∑

σ∈Kp

aσσ|aσ ∈ Zg;Vae

aσ 6= 0 =⇒ σt�b²|,`dt�X[t�|�g;~

}.

�l� `c\zbdX[`dtu|OX[tuV]o_XcZa\Kq�gimΦ∞ ea\O^�Va\,e$tuV X[Z]\{m]`c\z�Dt�kpwab�bc\z|zX[t�kpV��]��\{op\zX³g_q�gim

Φ∞ : Mp −→ CΦp (M,Z).

� � � � � � �¥:t��7g;V_kp`ct�\,V�X}g�X[tuk;Vfhskp` \,g;|}Z

p6 |,\,~�~

σg;V]eµtuea\zV>Xct�hj�

−σ��t�X[Z

σopt��p\zV_XcZa\�kpm]m�kpbdt�Xc\

k;`ctu\zV�X}giXctukpV��� c���� nqp �qr �M«O¬

σ�O«�,T©Oª±§j¬C§G©5,0/

p v ©}«w/j/7� j{ σ 6= σ§u¨µ©Oª±§j¬.§G©5,0/ � ¬s¯�«w.

< Φ∞(σ), σ >= 0.

� klql � � b�bc\,\zV�t�VTXcZa\Kma`ckDk;h:k;h:~u\,qnqyg % � � �

Φ∞(σ) = σ + c,

��Z]\,`c\c ∈ Im(V ) ⊆ M⊥

p

��

� �c l�kc�� nqp � n 1�¯�«#';,w) � � � � � §u¨�,0.�§u¨�®('�®�ª�)�¯D§u¨w';�� klql ���skpV�X[k ��vxw]mam�k;bc\

c ∈ CΦp (M,Z)

�3g;VaeT~u\OX

c =∑

σ ��� � ��� ����< c, σ > σ ∈ Mp.

�F\KbcZ�gi~u~�bd\,\KX[Z�giXΦ∞(c) = c

��¥�`dkpq ~�\,qnq�g % �  �]hskp`�g;V�� |z`ct�Xctu|,g;~�|z\,~u~σ

< Φ∞(c), σ >=< c, σ > .

vDkΦ∞(c) − c

tub�gΦ6Wt�V��;gi`ct�giV>X�|}Z�g;tuVTbdwa|}Z$XcZ�giX�hskp`�g;V�� |z`ct�Xctu|,g;~�|z\,~u~

σ

< Φ∞(c) − c, σ >= 0.

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� � ��������� ������������������������������ ���������!�#"%$

��� ~u\zqyqyg % � � � ��\{Zag��;\Φ∞(c) − c = 0

��  6  ��vxwam]m�kpbd\

c ∈ Mp

bcgiX[t�bd^�\zbΦ∞(c) = 0

�A´²Za\zV���hskp`�g;V�� |z`ct�Xctu|,g;~�|z\,~u~σ�

< Φ∞(c), σ >= 0.

�¡\�w]bc\ g;opg;tuV�XcZa\�\z½Dwag;~ut�XW�Φ∞(c) = c + cV

�²��Za\,`d\cV ∈ Im(V ) ⊆ M⊥

p

�´²Za\,`d\zhskp`d\;�ahskp`�g;V��T|z`ct�Xctu|,g;~�|z\,~u~

σ�

< c, σ >= 0,

��Zatu|}Ztuqnma~ut�\,bc = 0

��´²Z]\,kp`d\,q % � � tuqnma~ut�\,blX[Z�giX²XcZa\f�$kp`dbc\K|,kpqnma~�\O� t�b²tubdkpqyk;`cmaZat�|�X[k

M : 0 −→ Mn

e∂−→ Mn−1

e∂−→ . . .

e∂−→ M0 → 0

� � � � � � �¶�kpVabct�ea\,`²XcZa\�hskp~u~�k0��t�Vao7|zkpqnq(w]X}g�X[t��;\�eat�giop`[g;q

Mp

e∂−→ Mp−1

Φ∞ ↓ Φ∞ ↓

CΦp

∂−→ CΦ

p−1

�¡\fkp�]X}gituV��3hskp`c ∈ Mp

�(Φ∞)−1(∂Φ∞(c)) = ∂c

�B´²Z�g;Va�>b�Xck�X[Z]\7�8\,opt�VaVatuV]o�k;hX[Za\Kma`dkxk;h=k;h=X[Z]\,kp`d\,q % � � �]��\KZ�g��p\

∂c =∑

eσ∈Kp−1 and critical

< ∂Φ∞(c), σ > σ.

vxka�at�hc ∈ Mp

�σt�b�gµ|,`ct�X[t�|�g;~

(p− 1)6 hCg;|z\

< ∂c, σ >=< ∂Φ∞c, σ >=< Φ∞∂c, σ > .� � � � � � �

vxtuVa|z\H∗(M) ∼= H∗(M,Z)

�p��\³~u\�gi`cVµhs`ckpq�X[Za\��BV]t��p\z`cbcg;~�¶�kx\ �y|ztu\zV>Xl´²Za\zkp`c\zqX[Z�giX²hsk;`�g;V��#^�\,~�e

F�

H∗(M⊗ F) ∼= H∗(M) ⊗ F ∼= H∗(M,Z) ⊗ F ∼= H∗(M,F).

´²Z>wabM ⊗ F

t�b{g$eat���\z`c\,V�Xct�g;~ |zkpqnma~u\±�Yk;h��;\,|OX[kp`fbcm�g;|z\,bfk0�;\,`F��t�X[ZYX[Za\

b[g;qn\�Zakpqnkp~uk;o;�#gibM���kp`c\zk0�;\,`z�

dimFMp ⊗ F = mp(f)�at�X²hskp~u~�k���b�X[Z�g�X

� l�kl��"�#$&%�nqp � ' m{M§u¨ ,�� .�§j¬W«%& -�©}®('�)+/u«Sx � f ,y¦;§u¨�©Oª[«O¬W«#,n¦;§u¨�©Oª[«O¬ «³­F®�ª[¨�«�q*�.�©O¬.§G®(.®(.

M,0.8¦

F§u¨ ,0.�°µ©}®�«m~Á©O§G«w.�¬���«w/�¦ � ¬j¯�«w.T¬s¯�« �3¬Cª[®(.�4{­F®�ª[¨�« w.�«���*�,0/ v§j¬.§G«}¨G,0.8¦n¯�«w.�©}«�¬s¯�« -n«5, �µ­¼®�ª[¨�« w.�«���*�,0/ §j¬C§G«}¨Q¯�®(/�¦e�

Page 37: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8���������� � ��� �������� � ����Y��� � ��������������� ��������� � �

$³k0� ��\�~ukDkp�µhskp`²g(bcw]t�X}gi�a~u\B\O�xma`c\zbcbdtukpVThskp`∂� �F\Q��tu~�~�bdZak�� XcZ�giX�hskp`

τ (p+1)giVaeσ(p) XW��k�|,`dt�Xctu|�gi~l|z\,~�~ub,� < ∂τ, σ >

|�g;V¡��\y\O�xma`c\zbcbd\,e£tuVFXc\,`dq k;h�4;ª5,�¦;§G«w.�¬)�,;¬j¯�¨Khs`ck;q

∂τXckσ�

b�cedgf�hjikhml f nqp�� �t 4;ªS,�¦;§G«w.�¬ )�,;¬j¯À®�{n¦;§j'�«w.]¨O§G®(.p§u¨;,¨�«���*3«w.�©}«

γ®�{p v ©}«w/j/ ¨®�{

Mγ = σ0, σ1, . . . , σr¨w*�©}¯¬j¯�,;¬+{�®�ª7«wy;«Oª±°

i = 0, . . . , r − 1 �| §�}À§ {

V (σi) = 0¬j¯�«w.

σi+1 = σi

�| §j§�}À§ {

V (σi) 6= 0¬j¯�«w.

σi+1 < V (σi),0.8¦

σi+1 6= σi

�-n«�¨�,;°

γ§u¨G, op`[gieatu\zV>X�magiX[Z;{±ª}®('

σ0

¬W®σr

� 1�¯�«(~�\,Vao;XcZ£®�{γ � ¦D«w.�®�¬W«c¦ |γ|§u¨µ«���*�,0/�¬W®

r�

´²Z>wab���\{Z�g��p\KX[Z]\KXW��k_hskp~u~�k0��t�Vao(ma`dkpm8\,`dXctu\zb� c���� nqp � � | §�} m{

γ = σ0, σ1, . . . , σr

§u¨�,�4;ªS,�¦;§G«w.�¬�)�,;¬s¯�¬j¯�«w.�{�®�ªY«5,D©}¯i = 0, . . . , r − 1

«O§j¬j¯�«Oªσi = σi+1

®�ªf(σi) > f(σi+1)

�| §j§�} m{

γ1 = σ0, σ1, . . . , σr

,0.8¦γ2 = σr+1, σ1, . . . , σr+s

,;ª[«_¬���®#¨�«���*3«w.�©}«}¨y®�{p v ©}«w/j/ ¨ � ¬j¯�«w.

σ0, σ1, . . . , σr+1, . . . , σr+s§u¨G,�4;ªS,�¦;§G«w.�¬@)�,;¬j¯§ {G,0.8¦T®(.z/ °n§ {γ1,0.8¦

γ2,;ª[«�4;ªS,�¦;§G«w.�¬q)�,;¬s¯�¨��

�F\fV]k0��tuV�X[`dkxe]wa|,\�X[Z]\fVakiX[tuk;Vkih*q(w]~�X[t�ma~ut�|,t�XW�ykih g_op`[gieatu\zV>X�magiX[ZX[k � qn\�g(6bdwa`c\ � XcZa\���g��Tkp`dtu\zV>X[giX[t�kpV tub²|,g;`c`dtu\ze$gi~ukpVao_gµop`cg;eat�\,V�X�m�giXcZ��vxwamam8kpbd\

σ 6= σgi`c\�XW��k

p6W|z\,~�~ub²k;h

Mg;V]e

τtub�g

(p+ 1)6 |,\z~u~���t�X[Z

σ < τg;V]e

σ < τg;V]e!��k;XcZ!g;`d\#`c\zopwa~�gi`fhCg;|z\,b(k;h

τ�Y´²Za\,V"g;V!k;`ctu\zV�X}giXctukpVFkpV

σtuV]eawa|,\zb

giVÀkp`ct�\,V�X}g�X[tuk;V¼k;VσtuV¼X[Za\nhskp~u~�k���tuVao ��g��;� � $�kp`dtu\zV>X[giX[t�kpV¼kpV

σt�Vaeawa|z\,b(giV

k;`ctu\zV�X}giXctukpVÀk;Vτbdk X[Z�giX

< ∂τ, σ >= −1� ��t��;\,VÀXcZa\Tkp`ct�\,V�X}giXctukpV¡kpV

τ�¸��\

|}Z]kxkpbd\yXcZa\�kp`dtu\zV>X[giX[t�kpVYkpVσbckX[Z�giX

< ∂τ, σ >= 1�T¹A½>wat��;gi~u\,V�Xc~��p�M^]�xtuV]o�g;V

k;`ctu\zV�X}giXctukpV kpVσg;Vae

τ�3giV$kp`dtu\zV>X[giX[t�kpV tub²t�Vaeawa|z\,ekpV

σbdk_X[Z�giX

< ∂τ, σ >< ∂τ, σ >= −1.

' Va\�|,g;V b[g��#XcZ�giX���\�tuV]eawa|,\KgiV kp`ct�\,V�X}giXctukpVykpV� � bd~ut�eatuVao �

σg;|z`ckpbdb

τX[kσ�

¥:tuopwa`d\ % � � � � m]`c\,bd\,V�X[b³X[Zat�b²ma`ckD|,\zbcbz�3��t�X[Zgµ^]�x\,ekp`ct�\,V�X}giXctukpV kihσ�

�$k;`c\,k��;\,`,�8t�hσ = σ

�aX[Z]\,V$XcZa\Kkp`ct�\,V�X}g�X[tuk;Vt�Vaeawa|z\,ekpVσq(w]bdX³��\KX[Z]\{bcg;qn\

k;`ctu\zV�X}giXctukpV_kpVσ� ´²Z>wabz�]t�h

γ = σ0, σ1, . . . , σr

tublgKop`[g;e]tu\,V�Xlm�giXcZ��σ0

tuV]eawa|,\zb�giVk;`ctu\zV�X}giXctukpV$�>��ea\,o;`c\,\zb�kpV

σi

g;V t�V�m�g;`�X[t�|,wa~ug;`Bk;Vσr

�fvxtuV]|,\7��\µZ�g��p\(^]�x\zeFgiV

Page 38: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

� � ��������� ������������������������������ ���������!�#"%$

¥*t�opwa`d\ � � � � � �AUWV]eawa|,\ze�kp`dtu\zV>X[giX[t�kpV

kp`ct�\,V�X}giXctukpVµhskp`¸\,g;|}Z#|,\z~u~3k;hM�x��\�bd\zX

m(γ)���

m(γ) = 1t�h�XcZa\³tuVaeaw]|,\,e�kp`ct�\,V86

X}giXctukpV���σ0

k;Vσr

t�b³XcZa\7k;`ctu\zV�X}giXctukpVTk;h XcZatub³kpVa\i��g;V]em(γ) = −1

kiX[Za\z`d��tubd\;�¹¸½>wat��ig;~u\zV>Xc~��;�]��\Kbd\zX

m(γ) =

r−1∏

i=0

V (σi)6=0

< ∂V (σi), σi+1 > .� � � � � � �

�¡\K|�giVw]bc\KX[Zat�b�hskp`cq(w]~�g7X[k_ea\O^�Va\KX[Za\Kq(w]~�X[t�ma~ut�|,t�XW�µk;hAgiV>�#op`[g;e]tu\,V�X�m�g�X[Z��bCcedgf�hjikhml+fonqp � ���M«O¬

γ = σ0, σ1, . . . , σr

�O«�,E4;ªS,�¦;§G«w.�¬ )�,;¬s¯"®�{�¦;§j'�«w.]¨O§G®(.p�

-n«f¦D«m� .�«K¬j¯�«(q(w]~�X[t�ma~ut�|,t�XW�F®�{γ � m(γ)

�,°n¬s¯�« {�®�ª '�*�/7, |�� � � � � }0�

�F\{Vak;X[\�XcZ�giX³X[Zat�b³V]k;X[t�kpVTtub�|,kpqnm�giXctu�a~�\���t�XcZXcZa\{|,kpqnm�k;bct�XctukpV kih op`cg;eatu\zV�Xm�giX[Z]b,���$kp`c\fma`c\z|,t�bc\,~��p�8t�h

γ0 = σ0, σ1, . . . , σr

g;Vaeγ1 = σr+1, . . . , σr+s

gi`c\fXW��kop`[g;e]tu\,V�X�m�g�X[Zab�k;h:b[g;qn\�eat�qy\zVabct�kpV��xXcZa\,V

m(γ1)m(γ0) = m(γ1 ◦ γ0)� � � � � � �

��Za\,`d\γ1 ◦ γ0 = σ0, . . . , σr, σr+1, . . . , σr+s

�¥3k;`

p6W|,\z~u~�b

σg;Vae

σ��~u\OX

Γr(σ, σ)X[Z]\7bc\OX�k;h¸gi~u~Mop`[gieatu\zV>XBm�giXcZabBhs`dkpq

σX[kσk;h

~u\,V]o;X[Zr�

�F\KVak��5bcZ]k0��X[Z�giX�t�hτ (p+1) giVae σ(p) g;`d\�XW��kn|,`dt�X[t�|�g;~�|,\z~u~ubz�]X[Z]\,V

< ∂τ, σ >=∑

eσ(p)<τ

< ∂τ, σ >∑

γ∈ΓN (eσ,σ)

m(γ)

hskp`�g;V��N~�gi`cop\Q\zVakpwao;Z��

Page 39: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

�8���������� � ��� �������� � ����Y��� � ��������������� ��������� � � #

b�cedgf�hjikhml f nqp�� ���(«m� .�«L,T`d\,eawa|z\,e �k��Φ : Cp(M,Z) → Cp(M,Z)

�,°

Φ = 1 + ∂V

´²Zak;waopZΦe]t���\,`cbKhs`ck;q

�:��\y��t�~u~ ma`dk��p\yXcZ�giX{��\y|�g;VF`c\zma~�g;|z\

Φ�>�

ΦtuV

\z½>w�giX[t�kpV � % � � � � �O�A´²Z]\{opg;tuVTtub�XcZ�giXΦtub²\�gibctu\z`²X[Z�g;V

ΦX[k_|,k;qymawxX[\;�

� c���� nqp � "��=®�ªL,0.�° ©OªO§j¬C§G©5,0/:©}«w/j/ ¨τ (p+1) ,0.8¦ σ(p)

< ∂τ, σ >=< Φ∞∂τ, σ > .

� klql �*U X³t�b²bcw��n|,t�\,V�X³X[k_ma`ck��;\KXcZ�giX�hskp`²\O�p\,`��r ≥ 0

< Φr∂τ, σ >=< Φr∂τ, σ > .

´²Z]tub²hskp~u~�k���b�hs`ckpq�X[Za\Kkp�]bc\,`��igiX[t�kpV X[Z�giX²hsk;`²\z�p\z`d�|}Z�g;t�Vcg;VaeT\z�;\,`d�

r ≥ 0

Φr(c) − Φr(c) ∈ Im(V ) ⊆ M⊥∗ .

�F\Kma`ck��;\{XcZatub²��� t�Vaeawa|OX[tuk;Vk;Vr�

U hr = 0

XcZa\,VTX[Za\z`c\Ktub²V]k;X[Zat�VaoµX[k_ma`dk0�;\;�vDwamam8kpbc\KX[Za\Kma`dkpm8\,`dXW�#tub�X[`dwa\�hskp`

r − 1�¸¥�kp`

r�

Φr(c) = Φ(Φr−1(c)) = Φ(Φr−1(c) + V (c))

hsk;`²bckpqn\�|}Z�g;t�Vc�s��� X[Za\KtuV]eawa|zXct��;\�Z>�Dm8k;X[Z]\,bct�b3�±�

Φr(c) = (Φ + V ∂)(Φr−1(c) + V (c))

= Φr(c) + Φ(V (c)) + V ∂Φr−1(c) + V ∂V (c)

= Φr(c) + V (c+ ∂Φr−1(c) + ∂V (c))

��Z]\,`c\KX[Z]\{~ug;bdX�\,½>w�g;~�t�XW�#hskp~�~uk���b�hs`dkpqΦV = V + ∂V 2 = V

��

� c���� nqp � %∀(σ

(p)1 , σ

(p)2 ) ∈ (Kp)

2 �

< Φσ1, σ2 >=∑

γ∈Γ1(σ1,σ2)

m(γ).� � � � � �

� klql �:�F\Keatub�X[t�Vaopwat�bcZ \,~�\,qn\,V�X}gi`d� |,g;bc\zb³Xcknm]`ck��p\KX[Zat�b,�¥:tu`cb�X��abdwamam8kpbc\

V (σ1) = 0�A´²Z]\,V

< Φσ1, σ2 >=< σ1, σ2 >=

{1, si σ1 = σ2

0, si σ1 6= σ2

Page 40: webusers.imj-prg.frchristian.blanchet/Articles/Ecole... · ... ZakDkp~3 gib X[k(\zbdX[g; a~ut bcZ# ]`ctue]op\,b \OXW \z\,VTeat \,`d\,V X ... eab, Mg;V]eYXcZa\,t `Qeat bc|,`d\zXc\µ

� % ��������� ������������������������������ ���������!�#"%$

' VTX[Z]\{kiX[Za\z`BZ�giVae��aX[Z]\fkpV]~��#o;`[g;eat�\,V�X³m�giXcZ$k;h*~u\zVao;X[Z  ��\zoptuV]VatuVaongiXσtub²XcZa\

X[`ct��Dt�g;~�k;Va\γ = σ1, σ1

t�hσ2 = σ1

g;Vae�bdkm(γ) = 1

��U hσ1 6= σ2

�Γ1(σ1, σ2) = ∅bck

γ∈Γ1(σ1,σ2)

m(γ) = 0.

�¡\KZ�g��p\�XcZa\K\,½>w�g;~�t�XW� �8\zXW��\,\zV$X[Z]\KXW��kn\O�xma`c\zbcbdtukpVabz�$Bk��K��bcwam]m�kpbd\

V (σ1) 6= 0� U h

σ1 = σ2

�p��\²|�g;~�|,wa~ugiX[\lX[Za\�~�\zhjX*Z�g;Vae_bct�ea\�k;h�� % � � � 0�X[kµ^�Vae

< Φσ1, σ1 >=< σ1, σ1 > + < ∂V (σ1), σ1 >= 1 − 1 = 0.

�¡\ |,g;~u|zwa~�giXc\�X[Z]\`dtuopZ�XµZag;Vae"bct�ea\ k;h%� % � � � �O�"vDtuVa|z\V (σ1) 6= 0

�AXcZa\,`d\t�bµVakop`[g;e]tu\,V�X�m�g�X[ZTk;h:~u\,V]o;X[Z Â hs`ckpq

σ1

Xckσ1

bckΓ1(σ1, σ1) = ∅

giVae∑

γ∈Γ1(σ1,σ1)

m(γ) = 0.

$Bk��K��bcwam]m�kpbd\σ1 6= σ2

X[Za\zV

< Φσ1, σ2 >=< σ1, σ2 > + < ∂V (σ1), σ2 >=< ∂V (σ1), σ2 > .

U hσ2

tubBVak;X�g_hCg;|,\fk;hV (σ1)

�8X[Za\zV< Φσ1, σ2 >= 0

�³UWV�X[Zat�bB|,g;bc\7XcZa\,`d\µg;`d\7Vakop`[g;e]tu\,V�X�m�g�X[Zab�k;h:~u\zVao;X[Z  hs`ck;q

σ1

Xckσ2

bdknXcZ�giX∑

γ∈Γ1(σ1,σ2)

m(γ) = 0.

U hσ2t�b³g_hCg;|z\fkih

V (σ1)��X[Z]\,V$XcZa\,`d\f\±�]g;|zXc~��TkpVa\{op`cg;eatu\zV�XBmagiX[Zk;h ~�\,VaoiX[ZThs`ckpq

σ1Xckσ2

�γ = σ1, σ2

��vxk

m(γ) =< ∂V (σ1), σ2 >

g;b²ea\,bdtu`d\,e���

� �c�l�kc�� n@p�" �∀(τ (p+1), σ(p)) ∈ (Kp+1 × Kp)

�*¯D§G©}¯�,;ª[«(©Oª±§j¬.§G©5,0/*©}«w/j/ ¨

< ∂τ, σ >=∑

eσ(p)<τ

< ∂τ, σ >∑

γ∈ΓN (eσ,σ)

m(γ)

{�®�ªN/7,;ª 4D«(«w.�®(*04�¯��

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N~ug;`co;\Q\,Vak;waopZ

< ∂τ, σ >=< ΦN∂τ, σ > .vDtuVa|z\

∂τ =∑

eσ(p)<τ < ∂τ, σ > σ�a�¡\�^�V]e

< ∂τ, σ >=∑

eσ(p)<τ

< ∂τ, σ >< ΦN σ, σ >

hsk;`N~�g;`dop\Q\,V]kpwaopZ��A�F\Kma`ck��p\{��� tuV]eawa|zXctukpVTkpV

r ≥ 1

< Φrσ, σ >=∑

γ∈Γr(eσ,σ)

m(γ).

´²Z]\{|,g;bc\r = 1

t�b²~u\zqyqyg % � # �vDwamam8kpbc\KX[Za\Kma`dkpm8\,`dXW�#tub�X[`dwa\�hskp`

r − 1�

< Φrσ, σ > = < Φ(Φr−1σ), σ >

=∑

σ′(p)

< Φr−1σ, σ′ >< Φσ′, σ > (Φr−1σ =∑

σ′(p)

< Φr−1σ, σ′ > σ′)

=∑

σ′

γ∈Γr−1(eσ,σ′)

m(γ) < Φσ′, σ >�C��� t�Vaeawa|OX[tuk;VZ��Dm8k;X[Za\zbct�b3�

=∑

σ′

γ∈Γr−1(eσ,σ′)

m(γ)∑

γ′∈Γ1(σ′,σ)

m(γ′)�C��� ~u\zqyqyg % � � �

=∑

γ∈Γr(eσ,σ)

m(γ)�C��� ~�\,qnq�g % � � �Ct�tj�lg;Vae � % � � � �e�5�

vDk_XcZa\Kma`ck;m�\z`dXW�Ttub�Xc`cwa\�hskp`rg;Vae X[Za\Kma`dkxkih*t�b²|,kpqnma~u\OX[\i�

9 :�� � KCKCJOL�D�P�JMLOI7RTS PfRCA K �V? S@K ��D�JML±N ��SOVP��VPSQ

h�([2^ G^ � !�U�a8W

´=`ctug;ea\�|z\,~u~�wa~�gitu`c\i�x` ���±gi`c`[giVaop\,qn\,V�X�e � g;Vabd\,bz�� �c l�kc�� nqp�"@r 1�®(*D¬

h v ©}®0�O®�ªd¦;§u¨w'�« &(� � ¨O§j'�)+/u«w'�«w.�¬�©}®(.z.�«Sxx« � «w.�¬.ª[«!y(,;ª±§���� ¬��� ¨¦D«7¦;§j'�«w.]¨O§G®(.T¨w*k)��� ª±§G«w*Dª[«(®(*���� 4e,0/u«}¨��� 5«}¨O¬�������*D§jy(,0/u«w.�¬��� ]*�.])8ª[®,¦0*x§j¬��

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� � ��������� ������������������������������ ���������!�#"%$

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