Conjugação de involuções e suas aplicações

Conjugação de involuções e suas aplicações

Conjugação de involuções e suas aplicações

Elizabeth Ruth Salazar Flores

Orientadora: Profa. Dra. Miriam Garcia Manoel

Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação - ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências - Matemática . VERSÃO REVISADA

USP – São Carlos

Julho de 2013

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

S159c

Salazar Flores, Elizabeth Ruth

Conjugação de involuções e suas aplicações / Elizabeth Ruth Salazar Flores; orientadora Miriam Garcia Manoel. -- São Carlos, 2013.

74 p.

Dissertação (Mestrado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2013.

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♣♦r s❡r ♠❡✉ r❡❢ú❣✐♦ ❡ ♣♦r t❡r ❞❛❞♦ ❛ ♠✐♠ ❢♦rç❛s ♣❛r❛ ❛❧❝❛♥ç❛r ♠❡✉s ♦❜❥❡t✐✈♦s✳

❆♦s ♠❡✉s ❛♠❛❞♦s ♣❛✐s✱ ♣♦r t❡r ❛❝r❡❞✐t❛❞♦ ❡♠ ♠✐♠ ❡ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧✳ ❆♦s ♠❡✉s ✐r♠ã♦s ❘❛ú❧✱ ❆✉❣✉st♦✱ ▼✐❣✉❡❧ ❆♥❣❡❧✱ ❡ às ♠✐♥❤❛s ✐r♠ãs ❈♦t② ❡ ❍❛②❞❡é s♦✉ ❣r❛t❛ ♣♦r ♠✉✐t♦s ♠♦t✐✈♦s✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ ▼✐r✐❛♠ ▼❛♥♦❡❧✱ ♣❡❧♦ t❡♠❛ ♣r♦♣♦st♦ ❡ ♣♦r t❡r ❛❝❡✐t❛❞♦ ♠❡ ♦r✐✲ ❡♥t❛r s❡♠ ♠❡ ❝♦♥❤❡❝❡r✳ ❆ ❙♦❧❛♥❣❡✱ ♣❡❧❛ ❛❥✉❞❛ ❡ ♦ t❡♠♣♦ ♥❡st❡ tr❛❜❛❧❤♦✳

❆♦s ♠❡✉s ❝❛r♦s ❝♦❧❡❣❛s ❡ ❛♠✐❣♦s✱ ❏♦r❣❡ ❈✉r✐♣❛❝♦✱ ▲✐t♦✱ ❘♦s✐t❛✱ ❏❛q✉✐✱ P❛t❤②✱ ◆♦r❜✐❧✱ ■r✐s ❡ ❘❡♥❛t♦ ✱ s♦✉ ❣r❛t❛ ♣❡❧❛ ❛♠✐③❛❞❡✱ ❝♦♠♣❛♥❤✐❛ ❡ ♣❡❧❛ ❛❥✉❞❛ q✉❡✱ ❞❡r❛♠ s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐✳ ❆ ❈❧❛✉❞✐♦✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ♣❡❧❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❝♦♠❡ç❛r ♦ ♠❡str❛❞♦✳

➚ ❊❞✐s♦♥✱ ♣♦r t❡r ❡st❛❞♦ ❛♦ ♠❡✉ ❧❛❞♦ ♥❡ss❡s ú❧t✐♠♦s t❡♠♣♦s✱ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ❝♦♠✲ ♣r❡❡♥❞❡♥❞♦✱ ❛ss✐♠ ❝♦♠♦ ♣❡❧❛ s✉❛ ♣❛❝✐ê♥❝✐❛✱ ❝❛r✐♥❤♦ ❡ ❛t❡♥çã♦✳

➚ ❈◆Pq ♣❡❧♦ s✉♣♦rt❡ ✜♥❛❝❡✐r♦ ❝♦♥❝❡❞✐❞♦ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦✳

❊♥✜♠✱ à t♦❞♦s ❛q✉❡❧❡s q✉❡ ❝♦❧❛❜♦r❛r❛♠ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

❘❡s✉♠♦

❊st❡ tr❛❜❛❧❤♦ ♣r♦♣õ❡ ❞✐r❡❝✐♦♥❛r ♦ ❡st✉❞♦ ❞❡ ✐♥✈♦❧✉çõ❡s ♣❛r❛ ❞♦✐s r❛♠♦s ❞❡ ♣❡sq✉✐s❛ ❞❡♥tr♦ ❞❛ t❡♦r✐❛ ❞❛s ❙✐♥❣✉❧❛r✐❞❛❞❡s ❡ ❞❡ ❙✐st❡♠❛s ❉✐♥â♠✐❝♦s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ tr❛t❛✲ ♠♦s s✉❛ ✐♥t❡r❧✐❣❛çã♦ ❝♦♠ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❡ s❡✉ ❛♣❛r❡❝✐♠❡♥t♦ ♥♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❞✐s❝r❡t♦s r❡✈❡rsí✈❡✐s✳ ◆♦ ♣r✐♠❡✐r♦✱ tr❛t❛♠♦s ❞❛ ✐♠♣♦rt❛♥t❡ r❡❧❛çã♦ ❡♥tr❡ ❛ ❝❧❛s✲ s✐✜❝❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s✱ ❞✐❣❛♠♦s ❞❡ s ❞♦❜r❛s✱ ❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s ❛ss♦❝✐❛❞❛s ❛ ❡st❡s ❞✐❛❣r❛♠❛s✳ ◆♦ s❡❣✉♥❞♦ ❝♦♥t❡①t♦✱ ♦ ❡st✉❞♦ s❡ ✈♦❧t❛ ♣❛r❛ ❛ q✉❡stã♦ s♦❜r❡ ❝♦♥❞✐çõ❡s ♣❛r❛ ❛ ❧✐♥❡❛r✐③❛çã♦ s✐♠✉❧tâ♥❡❛ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ❛ ♦❜t❡♥çã♦ ❞❡ ❢♦r♠❛s ♥♦r♠❛✐s ❞❡ss❡s ♣❛r❡s✳

❆❜str❛❝t

❚❤✐s ✇♦r❦ ♣r♦♣♦s❡s t♦ ❛❞❞r❡ss t❤❡ st✉❞② ♦❢ ✐♥✈♦❧✉t✐♦♥s ❢♦r t✇♦ ❜r❛♥❝❤❡s ♦❢ r❡s❡❛r❝❤ ✐♥t♦ t❤❡ t❤❡♦r② ♦❢ ❙✐♥❣✉❧❛r✐t✐❡s ❛♥❞ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ tr❡❛t ✐ts ✐♥✲ t❡r❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ❞✐✈❡r❣❡♥t ❞✐❛❣r❛♠s ♦❢ ❢♦❧❞s ❛♥❞ t❤❡✐r ❛♣♣❡❛r❛♥❝❡ ✐♥ ❞✐s❝r❡t❡ r❡✈❡rs✐❜❧❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s✳ ❋✐rst✱ ✇❡ tr❡❛t t❤❡ ✐♠♣♦rt❛♥t r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❞✐✈❡r❣❡♥t ❞✐❛❣r❛♠s ♦❢ ❢♦❧❞s✱ s❛② s❢♦❧❞s✱ ❛♥❞ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ s✲t✉♣❧❡s ♦❢ ✐♥✈♦❧✉t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡s❡ ❞✐❛❣r❛♠s✳ ■♥ t❤❡ s❡❝♦♥❞ ❝♦♥t❡①t✱ t❤❡ st✉❞② t✉r♥s t♦ t❤❡ q✉❡s✲ t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥s ❢♦r s✐♠✉❧t❛♥❡♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❛ ❝❧❛ss ♦❢ ♣❛✐rs ♦❢ ✐♥✈♦❧✉t✐♦♥s ❛♥❞ t❤❡ ❞❡❞✉❝t✐♦♥ ♦❢ t❤❡ ♥♦r♠❛❧ ❢♦r♠s ♦❢ t❤❡s❡ ♣❛✐rs✳

❙✉♠ár✐♦

✶ Pr❡❧✐♠✐♥❛r❡s ✶

✶✳✶ ❉♦❜r❛s ❡ ✐♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ❈♦♥❥✉♥t♦s ❞❡ ✐♥✈♦❧✉çõ❡s tr❛♥s✈❡rs❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✸✳✶ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s tr❛♥s✈❡rs❛✐s ❝♦♠ ❞✐♠❡♥sã♦ ❞♦ s✉❜❡s♣❛ç♦ ❞❡ ♣♦♥t♦s ✜①♦s ❛r❜✐trár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳✷ ■♥✈♦❧✉çõ❡s ϕ ❝♦♠ codimF(ϕ) = 1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡

❞♦❜r❛s ✶✺

✷✳✶ ❈❛r❛❝t❡r✐③❛çã♦ ❞❡ ór❜✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❖ ❝❛s♦ s=n = 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷✳✶ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷✳✷ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ ❖ ❝❛s♦ s= 2✱ n≥3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✶ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✷ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸ ❉✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s✱ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ s✐st❡♠❛s ❞✐♥â✲

♠✐❝♦s ✸✶

✸✳✶ ❈❛♠♣♦s ✈❡t♦r✐❛✐s ❞❡s❝♦♥tí♥✉♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✷ ❉✐❢❡♦♠♦r✜s♠♦s r❡✈❡rsí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✶ ❈❛s♦ ♣❧❛♥❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✳✷✳✷ ❈❛s♦ n ≥3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✹ ●r✉♣♦s ❣❡r❛❞♦s ♣♦r ✉♠ ♣❛r ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ❛ t❡♦r✐❛ r❡✈❡rsí✈❡❧ ❡q✉✐✈❛✲

r✐❛♥t❡ ✸✾

✹✳✶ ●r✉♣♦s ❣❡r❛❞♦s ♣♦r ✉♠ ♣❛r ❞❡ ✐♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✶✳✶ ❈❛s♦ n = 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✶✳✷ ❈❛s♦ n ≥3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✷ ❆ t❡♦r✐❛ r❡✈❡rsí✈❡❧ ❡q✉✐✈❛r✐❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✷✳✶ ❚❡♦r✐❛ ❞❡ ❣r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✷✳✷ ●❡r♠❡s ❞❡ ❢✉♥çõ❡s ✐♥✈❛r✐❛♥t❡s ❡ ❞❡ ❛♣❧✐❝❛çõ❡s ❡q✉✐✈❛r✐❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✸ ●r✉♣♦s ❝♦♥❥✉❣❛❞♦s ❡ ❛ t❡♦r✐❛ r❡✈❡rsí✈❡❧ ❡q✉✐✈❛r✐❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✹ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✺ P❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❝♦♠ ❝♦♠♣♦st❛ ♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛ ✺✸ ✺✳✶ ❯♠ t❡♦r❡♠❛ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✷ ❈❧❛ss✐✜❝❛çã♦ ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s tr❛♥s✈❡rs❛✐s ❝♦♠ ❝♦♠♣♦st❛

♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✷✳✶ Pr✐♠❡✐r♦s r❡s✉❧t❛❞♦s s♦❜r❡ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✷✳✷ ❆ ❝❧❛ss✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✷✳✸ ❖ ❝❛s♦ ❤✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✺✳✷✳✹ ❖ ❝❛s♦ ❣❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✸

◆♦t❛çã♦

F(ϕ) : ❙✉❜❡s♣❛ç♦ ❞❡ ♣♦♥t♦s ✜①♦s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ϕ❀

Σ(f) : ❈♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s s✐♥❣✉❧❛r❡s ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦f❀

A(ϕ) : ❙✉❜❡s♣❛ç♦ ❛♥t✐♣♦❞❛❧ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ϕ❀

tr(ϕ) : ❚r❛ç♦ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ϕ❀

Rθ : ❘♦t❛çã♦ ❞❡ â♥❣✉❧♦ θ❀

Dm : ●r✉♣♦ ❞✐❡❞r❛❧ ❞❡ ♦r❞❡♠ m❀

PV(Γ) : ❆♥❡❧ ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s f :V →R Γ✲✐♥✈❛r✐❛♥t❡s❀

EV(Γ) : ❆♥❡❧ ❞♦s ❣❡r♠❡s ❞❡ ❢✉♥çõ❡s f : (V,0)→RΓ✲✐♥✈❛r✐❛♥t❡s❀

QV(Γ) : ▼ó❞✉❧♦ s♦❜r❡ PV(Γ) ❞❛s ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s f :V →R Γ✲❛♥t✐✲✐♥✈❛r✐❛♥t❡s❀

FV(Γ) : ▼ó❞✉❧♦ s♦❜r❡ EV(Γ) ❞♦s ❣❡r♠❡s ❞❡ ❢✉♥çõ❡s f : (V,0)→RΓ✲❛♥t✐✲✐♥✈❛r✐❛♥t❡s❀

~

PV,W(Γ) : ▼ó❞✉❧♦ s♦❜r❡ PV(Γ)❞❛s ❛♣❧✐❝❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s g :V →W Γ✲❡q✉✐✈❛r✐❛♥t❡s❀

~

EV,W(Γ) : ▼ó❞✉❧♦ s♦❜r❡ EV(Γ) ❞♦s ❣❡r♠❡s ❞❡ ❛♣❧✐❝❛çõ❡sg : (V,0)→W Γ✲❡q✉✐✈❛r✐❛♥t❡s❀

~

QV,W(Γ) : ▼ó❞✉❧♦ s♦❜r❡ PV(Γ) ❞❛s ❛♣❧✐❝❛çõ❡s ♣♦❧✐♥♦♠✐❛✐s g : V → W Γ✲r❡✈❡rsí✈❡✐s✲

❡q✉✐✈❛r✐❛♥t❡s❀

~

FV,W(Γ) : ▼ó❞✉❧♦ s♦❜r❡ EV(Γ) ❞♦s ❣❡r♠❡s ❞❡ ❛♣❧✐❝❛çõ❡s g : (V,0) → W Γ✲r❡✈❡rsí✈❡✐s✲

❡q✉✐✈❛r✐❛♥t❡s❀

■♥tr♦❞✉çã♦

❖ ❡st✉❞♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♥❡st❡ tr❛❜❛❧❤♦ é ❞❡ ♥❛t✉r❡③❛ ❧♦❝❛❧✱ ❝♦♠ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡ss❡ ❡♠ ❛❧❣✉♥s r❛♠♦s ❞❛ t❡♦r✐❛ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ❡ ❞♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s✳ ❆s ✐♥✈♦❧✉çõ❡s tê♠ s✐❞♦ ♦❜❥❡t♦ ❞❡ ✐♥t❡r❡ss❡ ❞❡ ❞✐✈❡rs♦s ❛✉t♦r❡s s♦❜ ❞✐❢❡r❡♥t❡s ❛s♣❡❝t♦s✳ ❙❡✉ tr❛t❛♠❡♥t♦ ♥❡st❡ ♣r♦❥❡t♦ é ❡♥❢❛t✐③❛❞♦ ❝♦♠ r❡s♣❡✐t♦ à s✉❛ ❛ss♦❝✐❛çã♦ ❝♦♠ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❡ ❝♦♠ ♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❞✐s❝r❡t♦s r❡✈❡rsí✈❡✐s✳

❖ ❡st✉❞♦ ♣r♦♣♦st♦ ❛❜♦r❞❛ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s s❡❣✉♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❛❞❛ ♣♦r ❝♦♥❥✉❣❛çã♦ s✐♠✉❧tâ♥❡❛ ❞❛s ✐♥✈♦❧✉çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡s✱ ♦✉ s❡❥❛✱ ❞✉❛ss✲✉♣❧❛s(ϕ1, . . . , ϕs)❡(ψ1, . . . , ψs)❞❡ ✐♥✈♦❧✉çõ❡s ❡♠(Rn,0)sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡

✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ h ❞❡ (Rn_,0)t❛❧ q✉❡

ψi =h◦ϕi◦h−1, i= 1, . . . , s.

Pr❡❧✐♠✐♥❛r♠❡♥t❡✱ sã♦ ❡st✉❞❛❞❛s ❡♠ ❞❡t❛❧❤❡s ❛s ❞❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞♦ ❛ss✉♥t♦✳ ❯♠❛ ✐♥✈♦❧✉çã♦ é ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ ϕ : (Rn_,0)→(Rn_,0)s❛t✐s❢❛③❡♥❞♦

ϕ◦ϕ = I✱ ♦♥❞❡I ❞❡♥♦t❛ ♦ ❣❡r♠❡ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ (Rn_,0)✳ ❚❡♠ ✉♠ ♣❛♣❡❧ ❝r✉❝✐❛❧ ❛q✉✐ ♦

❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ✜①♦s ❞❡ ϕ✱ ❞❛❞♦ ♣♦r

F(ϕ) = {x∈(Rn_{,0) :ϕ(x) = x}.}

❆♠❜♦s ♦s ❝♦♥t❡①t♦s ❡♠ q✉❡ ❛❜♦r❞❛♠♦s ❝♦♥❥✉♥t♦s ❞❡ ✐♥✈♦❧✉çõ❡s ♥❡st❡ tr❛❜❛❧❤♦ ❛ss✉✲ ♠❡♥ ❛ ❝♦♥❞✐çã♦ ❞❡ tr❛♥s✈❡rs❛❧✐❞❛❞❡✿ ✉♠ ❝♦♥❥✉♥t♦ {ϕ1, . . . , ϕs} ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ (Rn,0)✱

s ≤ n✱ é ❝❤❛♠❛❞♦ tr❛♥s✈❡rs❛❧ s❡ F(ϕi) é tr❛♥s✈❡rs❛❧ ❛ F(ϕj) ❡♠ 0 ♣❛r❛ i 6= j ❡

codim∩s

i=1T0F(ϕi) = Ps₁codimF(ϕi)✱ ♦♥❞❡ T0F(ϕi) ❞❡♥♦t❛ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ F(ϕi)

♥❛ ♦r✐❣❡♠✳

❆ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❡ ♥♦ss♦ ❡st✉❞♦ t❡♠ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❬✶✵❪ ❡ tr❛t❛ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s(f1, . . . , fs)❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡s✲✉♣❧❛s ❞❡

✐♥✈♦❧✉çõ❡s (ϕ1, . . . , ϕs) ❝♦♠ ϕi = I6 ❛ss♦❝✐❛❞❛s ❛ ❡st❡s ❞✐❛❣r❛♠❛s✱ ✐st♦ é✱ fi◦ϕi =fi ♣❛r❛

i = 1, . . . , s✳ ❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❡♠ ❬✶✵❪ é ✉♠❛ r❡❧❛çã♦ ✉♠✲❛✲✉♠ ❡♥tr❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ♦s ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞♦s ❛ ❡st❛s✳

❯♠❛ ê♥❢❛s❡ é ❞❛❞❛ ♣❛r❛ ❛ ❝❧❛ss✐✜❝❛çã♦ q✉❛♥❞♦s = 2 ❡ s♦❜ ❛ ❝♦♥❞✐çã♦ ❞❡ tr❛♥s✈❡rs❛✲ ❧✐❞❛❞❡ ❞♦s ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s✳ P❛r❛ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡st❡s ♣❛r❡s ♣r❡❝✐s❛♠♦s ✐♥❢♦r♠❛çã♦ ❞❛ ❡str✉t✉r❛ ❞♦ ❣r✉♣♦ ∆ = hϕ1, ϕ2i✱ ✐st♦ é✱ ❝♦♥❤❡❝❡r s❡ ♦ ❣r✉♣♦ ∆ é ❆❜❡❧✐❛♥♦ ♦✉ ♥ã♦✳

❊st❛ ❝❧❛ss✐✜❝❛çã♦ ♣❡r♠✐t❡ ❡♥❝♦♥tr❛r ✉♠❛ ✐♥✈❛r✐❛♥t❡ ♣❛r❛ ❡st❛s ❝❧❛ss❡s ❞❡ ❞✐❛❣r❛♠❛s ❞✐✲ ✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❛ q✉❛❧ é ♦ tr❛ç♦ ❞❛ ♣❛rt❡ ❧✐♥❡❛r ❞❛ ❝♦♠♣♦st❛ ϕ1◦ϕs✱ ❡ss❛ ✐♥✈❛r✐❛♥t❡

t❡♠ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ❞❛❞♦ q✉❡ ❛ ♠❛✐♦r✐❛ ❞❡ ❢♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s tr❛♥s✈❡rs❛✐s sã♦ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ❡st❛✳

❆❞✐❝✐♦♥❛♠♦s ❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❞❡s❝♦♥tí♥✉♦s ❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❡ s❡ ❛♣❧✐❝❛ ♦ ❡st✉❞♦ ❞❛ ❝❧❛ss✐✜❝❛çã♦ ❢❡✐t♦ ❛♦s s✐st❡♠❛s ❞✐s❝r❡t♦s r❡✈❡rsí✈❡✐s✳

◆♦t❡♠♦s q✉❡ s❡ ♦ ❣r✉♣♦ ∆ = hϕ1, ϕ2i é ❆❜❡❧✐❛♥♦✱ ❡♥tã♦ é ❞♦ t✐♣♦ Z2 ×Z2✳ ❆ss✐♠✱

❛♣❛r❡❝❡ ♦ ✐♥t❡r❡ss❡ ❞❡ ✐❞❡♥t✐✜❝❛r ❛ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞♦s ❣r✉♣♦s ❣❡r❛❞♦s ♣♦r ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s✳ ❉❡♣♦✐s ❞❡ ❢❛③❡r ❡st❛ ✐❞❡♥t✐✜❝❛çã♦✱ ✐♥tr♦❞✉③✐♠♦s ❛ t❡♦r✐❛ r❡✈❡rsí✈❡❧ ❡q✉✐✈❛r✐✲ ❛♥t❡ ♦♥❞❡ ✉s❛♠♦s ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❬✶✸❪ ❡ ❛♣❧✐❝❛♠♦s ❛ ❡st❡s ❣r✉♣♦s ❡♥❝♦♥tr❛❞♦s✳

❆ s❡❣✉♥❞❛ ♣❛rt❡ ❞❡st❡ ❡st✉❞♦ ❡♥✈♦❧✈❡ ❛ ❤✐♣❡r❜♦❧✐❝✐❞❛❞❡ ♥♦r♠❛❧ ❞❛ ❝♦♠♣♦st❛ ϕ1 ◦ϕ2

♣❛r❛ ♦ ♣❛r(ϕ1, ϕ2)❞❡ ✐♥✈♦❧✉çõ❡s ❡ t❡♠ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❜ás✐❝❛ ❬✶✶❪✳ ❉❛❞♦ q✉❡ ❛ ❧✐♥❡❛r✐③❛✲

çã♦ ❞❡ ϕ1 ❡ ϕ2 é ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ❧✐♥❡❛r✐③❛çã♦ ❞❡ ϕ1◦ϕ2 ✱ s✉r❣❡ ❛ q✉❡stã♦

s♦❜r❡ ❛ ❧✐♥❡❛r✐③❛çã♦ ❞♦ ♣❛r(ϕ1, ϕ2)✳ ❊♠ ❬✶✶❪ ♦s ❛✉t♦r❡s r❡s♦❧✈❡♠ ❡st❛ q✉❡stã♦ ♣❛r❛ ♣❛r❡s

❝✉❥❛ ❝♦♠♣♦st❛ é ♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✱ ❡ss❡ r❡s✉❧t❛❞♦ é ♠❡♥❝✐♦♥❛❞♦ ♥♦ ❚❡♦r❡♠❛ ✺✳✷✳ ❆ ❞❡✜♥✐çã♦ é ❛ s❡❣✉✐♥t❡✿ ❙❡❥❛ f : (Rn_{,0) → (}Rn_,0) ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦✱ _{f 6= I}✳

❙✉♣♦♥❤❛ F(f) ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❞❡ (Rn_,0) ❡ q✉❡ _{dimF(f) = k}✳ _f é ❝❤❛♠❛❞♦ ♥♦r♠❛❧✲

♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦ s❡ ♦ ❡s♣❡❝tr♦ ❞❡ df(0) t❡♠✱ ❝♦♥t❛♥❞♦ ♠✉❧t✐♣❧✐❝✐❞❛❞❡✱ n−k ❡❧❡♠❡♥t♦s ❢♦r❛ ❞♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦S1_{∩C✳ ❊st✉❞❛♠♦s ❛ ❝❧❛ss✐✜❝❛çã♦ ♣❛r❛ ❝❡rt♦s ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s}

❧✐♥❡❛r❡s ✐♥❝❧✉✐♥❞♦ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ ❛ ❝♦♠♣♦st❛ ϕ1◦ϕ2 é ❤✐♣❡r❜ó❧✐❝❛ ❡ ♥♦t❛♠♦s

q✉❡ s❡ ❞✉❛s ✐♥✈♦❧✉çõ❡s tê♠ ❝♦♠♣♦st❛ ♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✱ ❡♥tã♦ ♦ ❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ❡st❛s ✐♥✈♦❧✉çõ❡s ♥ã♦ é ❆❜❡❧✐❛♥♦✳

❊st❡ t❡①t♦ é ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥♦s ❈❛♣ít✉❧♦s ✶ ❡ ✷ ❛♣r❡s❡♥t❛♠♦s ❛ ❝❧❛s✲ s✐✜❝❛çã♦ ❞❛s ❢♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ❛s ❝♦rr❡s♣♦♥❞❡♥t❡s ❢♦r♠❛s ♥♦r♠❛✐s ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞♦s✳ ◆♦ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s ❛ ✐♥t❡r❛✲ çã♦ ❡♥tr❡ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❞❡s❝♦♥tí♥✉♦s ❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❡ ♦ ❡st✉❞♦ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s r❡✈❡rsí✈❡✐s ✉s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ♦❜t✐❞♦s ♥♦ ❈❛♣ít✉❧♦ ✷ ♣❛r❛ ♦ ❝❛s♦ ♣❧❛♥❛r ❡ ❧♦❣♦ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❛♦ ❝❛s♦ n✲❞✐♠❡♥s✐♦♥❛❧✳ ❆ r❡❢❡rê♥❝✐❛ ❜ás✐❝❛ ❞♦ ❡st✉❞♦ ❛♣r❡♥s❡♥t❛❞♦ ♥❡st❡s ❝❛♣ít✉❧♦s sã♦ ❬✶✵❪✳

◆♦ ❈❛♣ít✉❧♦ ✹ ❞❛♠♦s ❛ ❞❡s❝r✐çã♦ ❞♦s ❣r✉♣♦s ❣❡r❛❞♦s ♣♦r ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s

①✐✐✐

tr❛♥s✈❡rs❛✐s ♣r✐♠❡✐r❛♠❡♥t❡ ❡♠ ❞✐♠❡♥sã♦ 2 ❡ ❞❡♣♦✐s ❡♠ ❞✐♠❡♥sõ❡s ♠❛✐♦r❡s ✐❣✉❛✐s q✉❡ 3✳ ❋❛③❡♠♦s t❛♠❜é♠ ❛q✉✐ ✉♠ ❡st✉❞♦ s♦❜r❡ ❛ ❜❛s❡ ❞❛ t❡♦r✐❛ r❡✈❡rsí✈❡❧ ❡q✉✐✈❛r✐❛♥t❡ ♦ ❛♣❧✐❝❛♠♦s ❛ ❡st❡s ❣r✉♣♦s ❞❡ ✐♥✈♦❧✉çõ❡s ✳

◆♦ ❈❛♣ít✉❧♦ ✺ ♠❡♥❝✐♦♥❛♠♦s ✉♠ t❡♦r❡♠❛ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❝♦♠ ❝♦♠♣♦st❛ ♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✳ ◆❛ ❙❡çã♦ ✺✳✷ ❛♣r❡s❡♥t❛♠♦s ❛ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ♣❛r❡s ❧✐♥❡❛r❡s ✈✐❛ ❡q✉✐✈❛❧❡♥❝✐❛ ❧✐♥❡❛r ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ❝♦♠ ❝♦♠♣♦st❛ ❤✐♣❡r❜ó❧✐❝❛ ❞❡♥tr♦ ❞❛ ❝❧❛ss❡ ❞♦s ♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝♦s✳ ❆❞✐❝✐♦✲ ♥❛❧♠❡♥t❡✱ ❡①♣❧✐❝✐t❛♠♦s ❛ ❝❧❛ss✐✜❝❛çã♦ ♣❛r❛ ♣❛r❡s ❡♥ ❞✐♠❡♥sõ❡s ❡s♣❡❝í✜❝❛s✳ ❈♦♠♦ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ❞❡st❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s s✉❜✈❛r✐❡❞❛❞❡s F(ϕ1)✱ F(ϕ2) ❡

❈❛♣ít✉❧♦

1

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ♦s ♦❜❥❡t♦s ♣r✐♥❝✐♣❛✐s ❞❡st❛ ❞✐ss❡rt❛çã♦✱ q✉❡ sã♦ ✐♥✈♦❧✉çõ❡s ❡ ❞♦❜r❛s✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡♥tr❡ ❡❧❛s ❡ ♦s r❡s✉❧t❛❞♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s✳ ❈♦♠♦ ✈❡r❡♠♦s✱ t❛❧ ❝❧❛ss✐✜❝❛çã♦ ❡stá ✐♥t✐♠❛♠❡♥t❡ ❧✐❣❛❞❛ à ❝❧❛ss✐✜❝❛çã♦ ❞❛s s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s ❛ss♦❝✐❛❞❛s✳ ◆♦ss❛ r❡❢❡rê♥❝✐❛ ❜ás✐❝❛ é ❬✶✵❪✳

❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✜①♦s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ϕ✱ ❞❡♥♦t❛❞♦ ♣♦rF(ϕ)✱ ❡ ✐♥tr♦✲ ❞✉③✐♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ tr❛♥✈❡rs❛❧✐❞❛❞❡ ❞❡ ✐♥✈♦❧✉çõ❡s✳ ❉❡s❝r❡✈❡♠♦s ❛s ❢♦r♠❛s ♣r❡♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s tr❛♥s✈❡rs❛✐s ❝♦♠ codimF(ϕ) ❛r❜✐trár✐❛ ❡ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ❞♦✐s ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❞❡st❡ t✐♣♦ s❡❥❛♠ ❡q✉✐✈❛❧❡♥t❡s✳

❆♣r❡s❡♥t❛♠♦s ❛s ❢♦r♠❛s ♣r❡♥♦r♠❛✐s ❞❡ ✉♠❛s✲✉♣❧❛ ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s tr❛♥s✈❡rs❛✐s ❡ ❛ss✐♠ ❞❡s❝r❡✈❡♠♦s ❛s ❢♦r♠❛s ♣r❡♥♦r♠❛✐s ❞♦s ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞♦s ❛ ❡st❛s s✲✉♣❧❛s✳

❋✐♥❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❡①♣❧✐❝✐t❛♠❡♥t❡ ❛ ❢♦r♠❛ ♥♦r♠❛❧ ❞❡ ✉♠❛s✲✉♣❧❛ ❞❡ ✐♥✈♦❧✉çõ❡s tr❛♥s✈❡rs❛✐s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♥❡❛r❡s t❛✐s q✉❡ ♦ ❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ❡st❛s ✐♥✈♦❧✉çõ❡s é ❆❜❡❧✐❛♥♦✳ ❊♠ ❝♦♥s❡qüê♥❝✐❛✱ ♦❜t❡♠♦s t❛♠❜é♠ ❛s ❢♦r♠❛s ♥♦r♠❛✐s ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥✲ t❡s ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞♦s ❛ ❡st❛ s✲✉♣❧❛ ❞❡ ✐♥✈♦❧✉çõ❡s✳

✶✳✶ ❉♦❜r❛s ❡ ✐♥✈♦❧✉çõ❡s

❖s ❝♦♥❝❡✐t♦s ❞❡ ❞♦❜r❛s ❡ ✐♥✈♦❧✉çõ❡s tr❛t❛❞♦s ❛q✉✐ sã♦ ❞❡ ♥❛t✉r❡③❛ ❧♦❝❛❧ ❡ ❞❛♠♦s s✉❛s ❞❡✜♥✐çõ❡s ✉s❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ❣❡r♠❡✱ ❞❡✜♥✐çã♦ ❜ás✐❝❛ ❞❛ t❡♦r✐❛ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s✳

❉❡✜♥✐çã♦ ✶✳✶✿ ❯♠❛ ✐♥✈♦❧✉çã♦ é ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ ϕ : (Rn_{,0) → (}Rn_,0)

✷ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

s❛t✐s❢❛③❡♥❞♦ ϕ◦ϕ= I✳

❱❛♠♦s ❞❡♥♦t❛r ♣♦rF(ϕ) ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ✜①♦s ❞❡ ϕ✱ ✐st♦ é✱

F(ϕ) = {x∈(Rn_{,0) :ϕ(x) = x}.}

❉❡✜♥✐çã♦ ✶✳✷✿ ❉✉❛s s✲✉♣❧❛s (ϕ1. . . ϕs) ❡ (ψ1. . . ψs)❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ (Rn,0) sã♦ ❡q✉✐✲

✈❛❧❡♥t❡s s❡ ❡①✐st❡ ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ h ❞❡(Rn_,0) t❛❧ q✉❡_{ψi =h◦ϕi◦h}−1✱ ♣❛r❛

t♦❞♦ i= 1, . . . , s.

❖ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦h❞❛ ❉❡✜♥✐çã♦ ✶✳✷ é ♥❛❞❛ ♠❛✐s q✉❡ ✉♠❛ ❝♦♥❥✉❣❛çã♦ s✐♠✉❧✲ tâ♥❡❛ ❞❛s ✐♥✈♦❧✉çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❛s s✲✉♣❧❛s ✳ ◆♦t❡♠♦s q✉❡ ❡st❡ h s❛t✐s❢❛③

h(F(ϕi)) =F(ψi), i= 1, . . . , s. ✭✶✳✶✮

❉❡ ❢❛t♦✱ s❡x∈ F(ψi)t❡♠♦s q✉❡✱ψi(x) = x=h◦ϕi(h−1(x))♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡h−1(x)∈

F(ϕi) ❡✱ ♣♦rt❛♥t♦✱x∈h(F(ϕi))✳ ❙❡ x∈ F(ϕi)✱ t❡♠♦s q✉❡ψi(h(x)) =h(ϕi(x)) =h(x) ❡✱

♣♦rt❛♥t♦✱ h(x)∈ F(ψi)✳

❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦Gs ={ϕ1. . . ϕs}❞❡ ✐♥✈♦❧✉çõ❡s ❡♠(Rn,0)❡∆s =hϕ1. . . ϕsi

♦ ❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ❡st❛s ✐♥✈♦❧✉çõ❡s✳ ❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❝❤♥❡r✲ ▼♦♥t❣♦♠❡r② ❡♠ ❬✶✷❪✿

❚❡♦r❡♠❛ ✶✳✸✿ ❙❡ ∆s é ✉♠ ❣r✉♣♦ ❆❜❡❧✐❛♥♦✱ ❡♥tã♦ ❛ s✲✉♣❧❛ (ϕ1. . . ϕs) ❞❡ ✐♥✈♦❧✉çõ❡s é

❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ s✲✉♣❧❛ ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s✳

❉♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r t❡♠♦s q✉❡ q✉❛❧q✉❡r ✐♥✈♦❧✉çã♦ϕ : (Rn_,0)→(Rn_,0) é ❝♦♥❥✉❣❛❞❛

❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❧✐♥❡❛r✳ P❛r❛ ✉♠❛ ú♥✐❝❛ ✐♥✈♦❧✉çã♦ ♣♦❞❡♠♦s✱ ❛❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ❡①♣❧✐❝✐t❛r ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ q✉❡ ❝♦♥❥✉❣❛ ϕ ❡ s✉❛ ❞✐❢❡r❡♥❝✐❛❧ dϕ(0) ♥❛ ♦r✐❣❡♠✿

h= 1

2(I +dϕ(0)◦ϕ).

❙❡❣✉❡✲s❡ ❞❡ ✭✶✳✶✮ q✉❡ F(ϕ) é ❧♦❝❛❧♠❡♥t❡ ❞✐❢❡♦♠♦r❢♦ ❛ ✉♠ s✉❜❡s♣❛ç♦ ❧✐♥❡❛r ❞❡ Rn✳

P♦rt❛♥t♦✱ F(ϕ)é ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ❡♠ (Rn_,0)✳ ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝❛r❛❝t❡r✐③❛r ✉♠❛

✐♥✈♦❧✉çã♦ ϕ ❛ tr❛✈és ❞❛ ❞✐♠❡♥sã♦ ❞❡ F(ϕ)✿

✶✳ s❡ codimF(ϕ) = 0✳ ❊♥tã♦✱ ϕ é ❛ ✐❞❡♥t✐❞❛❞❡I❀

✷✳ s❡ codimF(ϕ) = l6= 0✱ ❡♥tã♦ ϕ é ❝♦♥❥✉❣❛❞❛ à ❢♦r♠❛ ❝❛♥ô♥✐❝❛

❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✸

❖✉tr♦ ❝♦♥❝❡✐t♦ ❜ás✐❝♦ ♣❛r❛ ♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s é✿

❉❡✜♥✐çã♦ ✶✳✹✿ ❯♠ ❣❡r♠❡ f : (Rn_{,0) → (}Rn_,0) é ✉♠❛ ❞♦❜r❛ s❡ ❡①✐st❡♠ ❣❡r♠❡s ❞❡

❞✐❢❡♦♠♦r✜s♠♦s h ❡ k ❞❡ (Rn_,0)t❛❧ q✉❡ _{f =k◦f}0_◦h−1✱ ♦♥❞❡

f0(x1, . . . , xn) = (x21, x2. . . , xn).

❆❣♦r❛ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ q✉❡ r❡❧❛❝✐♦♥❛ ✉♠❛ ❞♦❜r❛ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦✿

❉❡✜♥✐çã♦ ✶✳✺✿ ❉❛❞❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ϕ ❡♠ (Rn_,0) ❡ ✉♠❛ ❞♦❜r❛ _{f : (}Rn_{,0) 7→ (}Rn_,0)

❞✐③❡♠♦s q✉❡ f é ❛ss♦❝✐❛❞♦ ❛ ϕ✱ ♦✉ ϕ é ❛ss♦❝✐❛❞♦ ❛ f✱ s❡ ϕ6= I ❡ f◦ϕ =f✳

❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❛t✐✈♦s ❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡ à ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ❡st❛✳

Pr♦♣♦s✐çã♦ ✶✳✻✿ ❉❛❞❛ ✉♠❛ ❞♦❜r❛ f : (Rn_{,0) 7→ (}Rn_,0)✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ✐♥✈♦❧✉çã♦

❛ss♦❝✐❛❞❛ ❛ f✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ f0 _{❛ ❞♦❜r❛ t❛❧ q✉❡ f}0_(x

1, . . . , xn) = (x21, x2, . . . , xn)✳ ➱ ❢á❝✐❧ ✈❡r

q✉❡ f0 _{❡st❛ ❛ss♦❝✐❛❞❛ à ✐♥✈♦❧✉çã♦ ϕ}0_(x

1, . . . , xn) = (−x1, . . . , xn)✳ ❙❡❥❛ ϕ ♦✉tr❛ ✐♥✈♦❧✉çã♦

❛ss♦❝✐❛❞❛ ❛ f0_{✱ ♦♥❞❡ ϕ= (ϕ}

1, . . . , ϕs)✱ ✐st♦ é✱

f0 = (x2₁, x2, . . . , xn) = f0◦ϕ =f0(ϕ1, ϕ2, . . . , ϕn) = (ϕ21, ϕ2, . . . , ϕn).

❉✐st♦ t❡♠♦s ϕ2

1 = x21 ❡ ϕi = xi ♣❛r❛ i = 2, . . . , n✳ ❈♦♠♦ ϕ 6= I t❡♠♦s q✉❡ ϕ1 = −x1

❡✱ ♣♦rt❛♥t♦✱ ϕ = ϕ0_{✳ ❆❣♦r❛✱ s❡❥❛ f ✉♠❛ ❞♦❜r❛ q✉❛❧q✉❡r✳ P❡❧❛ ❞❡✜♥✐çã♦✱ s❛❜❡♠♦s q✉❡}

❡①✐st❡♠ k ❡ h ❣❡r♠❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s t❛❧ q✉❡ f =k◦f0_◦h−1_{✳ ❉❡✜♥✐♥❞♦ ❛ ✐♥✈♦❧✉çã♦}

ϕ ♣♦r

ϕ=h◦ϕ0◦h−1,

♦❜s❡r✈❛♠♦s q✉❡ f é ❛ss♦❝✐❛❞❛ ❛ ϕ✱ ✐st♦ é✱ f ◦ϕ = f✳ ❙❡❥❛ ψ ♦✉tr❛ ✐♥✈♦❧✉çã♦✳ ❊♥tã♦ f ◦ϕ=f◦ψ =f✳ ▲♦❣♦✱

f0◦h−1◦ψ◦h=k−1◦k◦f0 ◦h−1 ◦ψ◦h

=k−1◦f ◦ψ◦h

=k−1◦f ◦h

=f0,

♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ h−1 _{◦ψ ◦h é ✉♠❛ ✐♥✈♦❧✉çã♦ ❛ss♦❝✐❛❞❛ ❛ f}0_{✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ϕ}0_✱

✹ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

P❛r❛ ❛ ✐♥✈♦❧✉çã♦ϕ0 _{❞❛❞❛ ♥❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✻ t❡♠♦scodimF(ϕ}0_{) = 1✳ ❙❡❣✉❡✱}

❡♥tã♦✱ q✉❡ ♣❛r❛ q✉❛❧q✉❡r ❞♦❜r❛ f ❛ss♦❝✐❛❞❛ ϕ✱ t❡♠♦s

codimF(ϕ) = codimF(ϕ0) = 1,

s❡♥❞♦ ϕ ❡ ϕ0 _{❝♦♥❥✉❣❛❞❛s✳ ❆❧é♠ ❞✐ss♦✱ ♥♦t❡♠♦s q✉❡}

F(ϕ) = Σ(f),

♦♥❞❡ Σ(f) ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s s✐♥❣✉❧❛r❡s ❞❡ f✳ ❉❡ ❢❛t♦✱ ♥♦t❡♠♦s q✉❡F(ϕ0) =

{(0, x2, . . . , xn)∈(Rn,0)}✱

(df0)(x1, . . . , xn) =

    

2x1 0 . . . 0

0 1 ✳✳✳

✳✳✳ ✳✳✳ 0 0 . . . 0 1

    .

▲♦❣♦ (x1, . . . , xn) é ♣♦♥t♦ s✐♥❣✉❧❛r ❞❡ f s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x1 = 0✳ ❋✐♥❛❧♠❡♥t❡✱ ❣❡♥❡r❛❧✐③❛✲

♠♦s ❡st❡ ❢❛t♦ ♣❛r❛ q✉❛❧q✉❡r ❞♦❜r❛ f ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ✐♥✈♦❧✉çã♦ϕ✿ t❡♠♦s

f =k◦f0◦h−1 ❡ ϕ=h◦ϕ0◦h−1,

❝♦♠ k ❡ h ❣❡r♠❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s✳ ▲♦❣♦

df(x) =dk(f0◦h−1(x))◦d(f0)(h−1(x))◦(dh)−1(x) ❡ F(ϕ) = h(F(ϕ0)).

❖❜s❡r✈❛♠♦s q✉❡ x∈Σ(f)s❡✱ é s♦♠❡♥t❡ s❡✱h−1_(x)∈Σ(f0_{) = F(ϕ}0_{)✳ P♦rt❛♥t♦✱x∈Σ(f)}

s❡✱ é s♦♠❡♥t❡ s❡✱ x∈h(F(ϕ0_{)) =F(ϕ)✳}

Pr♦♣♦s✐çã♦ ✶✳✼✿ ❉❛❞❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ϕ ❡♠ (Rn_,0) ❝♦♠ _{codimF(ϕ) = 1}✱ ❡①✐st❡ ✉♠❛

❞♦❜r❛ f : (Rn_,0)7→(Rn_,0)✱ ❛ss♦❝✐❛❞❛ ❛ _ϕ✳

❉❡♠♦♥str❛çã♦✿ ❙❡ codimF(ϕ) = 1✱ ϕ é ❝♦♥❥✉❣❛❞❛ à ✐♥✈♦❧✉çã♦ ϕ0_(x

1, . . . , xn) =

(−x1, x2, . . . , xn)✱ ✐st♦ é✱ ❡①✐st❡ h ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ t❛❧ q✉❡ ϕ = h ◦ ϕ0 ◦ h−1✳

❉❡✜♥✐♥❞♦ ❛ ❞♦❜r❛ f =f0_◦h−1_{✱ t❡♠♦s f ❛ss♦❝✐❛❞❛ ❛ ϕ✳}

❖❜s❡r✈❛çã♦ ✶✳✽✿ ❆ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ✐♥✈♦❧✉çã♦ϕ♥ã♦ é ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛✳ ❉❡ ❢❛t♦✱ s❡ f é ✉♠❛ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ϕ✱ ❡♥tã♦ q✉❛❧q✉❡r g ∈ L ·f é t❛♠❜é♠ ❛ss♦❝✐❛❞❛ ❛ ϕ✱ ♦♥❞❡L é ♦ ❣r✉♣♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s à ❡sq✉❡r❞❛✳

❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✺

tr❛t❛♠♦s ❞❡ ✐♥✈♦❧✉çõ❡s ❛ss♦❝✐❛❞❛s ❛ ❞♦❜r❛s✳

❙❡❥❛i✉♠ ✐♥t❡✐r♦ ✜①♦✱ 1≤i≤n✳ ❈♦♥s✐❞❡r❡ ❛ ✐♥✈♦❧✉çã♦ ϕ0

i ❞❛❞❛ ♣♦rϕ0i(x1, . . . , xn) =

(x1, . . . ,−xi, . . . , xn) ❡ ❛ ❞♦❜r❛ fi0 = (x1, . . . , x2i, . . . , xn)✱ ❛ss♦❝✐❛❞♦ ❛ ϕ0i✳ ❊♥tã♦✿

▲❡♠❛ ✶✳✾✿ ❯♠❛ ❞♦❜r❛ g : (Rn_{,0) → (}Rn_,0) é ❛ss♦❝✐❛❞❛ ❛ _ϕ0

i s❡✱ ❡ só s❡✱ g é L✲

❡q✉✐✈❛❧❡♥t❡ ❛ f0

i✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ g é ❛ss♦❝✐❛❞❛ ❛ ϕ0

i✳ ❊♥tã♦

g◦ϕ0_i(x1, . . . , xn) = g(x1, . . . , xn)

g(x1, . . . ,−xi, . . . , xn) = g(x1, . . . , xi, . . . , xn).

❉❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣♦❞❡♠♦s ✈❡r q✉❡ g ♥ã♦ ❞❡♣❡♥❞❡ ❞♦ s✐♥❛❧ ❞❡ xi✱ ❛ss✐♠ ♣♦❞❡♠♦s

❡s❝r❡✈❡r g ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

g(x1, . . . , xi, . . . , xn) = (k1(x1, . . . , xi2, . . . , xn), . . . , kn(x1, . . . , x2i, . . . , xn))

= (k1◦f0, . . . , kn◦f0) =k◦f0,

♦♥❞❡ k = (k1, . . . , kn)✳ ❈♦♠♦ g é ✉♠❛ ❞♦❜r❛✱ ❡♥tã♦ k é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳ P♦rt❛♥t♦✱ g

é L✲❡q✉✐✈❛❧❡♥t❡ ❛ f0

i. ❘❡❝✐♣r♦❝❛♠❡♥t❡ s❡✱ g é ✉♠❛ ❞♦❜r❛ L✲❡q✉✐✈❛❧❡♥t❡ ❛ fi0✱ ❡①✐st❡ ✉♠

❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ k t❛❧ q✉❡ g =k◦f0

i✳ ▲♦❣♦

g◦ϕ0_i =k◦f_i0◦ϕ0_i =k◦f_i0 =g.

❆ss✐♠ g é ✉♠❛ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ϕ0_✳

❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ❣❡♥❡r❛❧✐③❛ ♦ ▲❡♠❛ ✶✳✾ ❛❝✐♠❛✿

Pr♦♣♦s✐çã♦ ✶✳✶✵✿ ❙❡❥❛ ϕ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠(Rn_,0)❡ s❡❥❛ _h✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦

❞❡ (Rn_,0) t❛❧ q✉❡ _{ϕ = h◦ϕ}0

i ◦h−1✳ ❈♦♥s✐❞❡r❡ ❛ ❞♦❜r❛ fi0◦ h−1 ❛ss♦❝✐❛❞❛ ❛ ϕ✳ ❊♥tã♦✱

❛ ❞♦❜r❛ g : (Rn_{,0) → (}Rn_,0) é t❛♠❜é♠ ❛ss♦❝✐❛❞❛ ❛ _ϕ s❡✱ ❡ só s❡✱ _g é _L✲❡q✉✐✈❛❧❡♥t❡ ❛

f0

i ◦h−1✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛g ✉♠❛ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ϕ✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ g◦h é ✉♠❛ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ϕ0

i✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ g◦h é ❡q✉✐✈❛❧❡♥t❡ ❛ fi0✱ ♦ q✉❛❧ q✉❡r ❞✐③❡r q✉❡ ❡①✐st❡

✉♠ ❞✐❢❡♦♠♦r✜s♠♦ k t❛❧ q✉❡

g◦h=k◦f0

i,

❧♦❣♦✱ g =k◦f0

i ◦h−1✬✳ ❆ss✐♠✱ g é L✲❡q✉✐✈❛❧❡♥t❡ ❛ fi0◦h✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s

g L✲ ❡q✉✐✈❛❧❡♥t❡ ❛f0

i ◦h−1✳ ❊♥tã♦✱ ❡①✐st❡k ❞✐❢❡♦♠♦r✜s♠♦ t❛❧ q✉❡g =k◦fi0◦h−1 ❡

✻ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

♦ q✉❛❧ t❡r♠✐♥❛ ❝♦♠ ❛ ♣r♦✈❛ ❞❡st❛ ♣r♦♣♦s✐çã♦✳

▼❛✐s ❣❡r❛❧♠❡♥t❡ ❛✐♥❞❛✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r s✉❜st✐t✉✐♥❞♦f0

i◦h−1

♣♦r ✉♠❛ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ϕ✿

❈♦r♦❧ár✐♦ ✶✳✶✶✿ ❙❡❥❛ ϕ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ (Rn_,0) ❡ s❡❥❛ _{f : (}Rn_{,0) → (}Rn_,0) ✉♠❛

❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ϕ✳ ❊♥tã♦ ❛ ❞♦❜r❛ g : (Rn_{,0)→ (}Rn_,0) é t❛♠❜❡♠ ❛ss♦❝✐❛❞❛ ❛ _ϕ s❡✱ ❡

só s❡✱ g é L✲❡q✉✐✈❛❧❡♥t❡ ❛ f✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ g L✲❡q✉✐✈❛❧❡♥t❡ ❛ f✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦k t❛❧ q✉❡g =k◦f✳ ❉✐st♦ t❡♠♦s q✉❡✱ g◦ϕ=k◦f◦ϕ=k◦f =g ❡✱ ♣♦rt❛♥t♦✱g é ❛ss♦❝✐❛❞❛ ❛ϕ✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡g é ❛ss♦❝✐❛❞❛ ❛ϕ✳ ❈♦♠♦ ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ϕ=h◦ϕ0

i ◦h−1

❡ ❞❡s❞❡ q✉❡f é t❛♠❜é♠ ❛ss♦❝✐❛❞❛ ❛ϕ✱ t❡♠♦s q✉❡f ❡g sã♦L✲❡q✉✐✈❛❧❡♥t❡s ❛f0

i ◦h−1✱ ✐st♦

é✱ ❡①✐st❡♠ ❣❡r♠❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s k1 ❡k2 t❛✐s q✉❡g =k1◦fi0◦h−1 ❡f =k2◦fi0◦h−1✳

▲♦❣♦✱

g =k1◦k−21◦k2 ◦fi0◦h−1 =k◦f.

❖♥❞❡ k =k1◦k2−1 ❡ ♣♦rt❛♥t♦✱ g éL✲❡q✉✐✈❛❧❡♥t❡ ❛f✳

✶✳✷ ❉✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s

(f1, . . . , fs) : (Rn,0)→(Rn×. . .×Rn,0)

é ❝❤❛♠❛❞♦ ❞✐❛❣r❛♠❛ ❞✐✈❡r❣❡♥t❡✳ ◆♦ ❡s♣❛ç♦ ❞❡st❡s ❞✐❛❣r❛♠❛s✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ é ❞❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿

❉❡✜♥✐çã♦ ✶✳✶✷✿ ❉♦✐s ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s

(f1, . . . , fs) : (Rn,0)→(Rn×. . .×Rn,0) ❡ (g1, . . . , gs) : (Rn,0)→(Rn×. . .×Rn,0)

sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡♠ ❣❡r♠❡s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ h, k1, . . . , ks ❞❡(Rn,0) t❛✐s q✉❡gi =

ki◦fi◦h−1✱ ♣❛r❛ t♦❞♦ i= 1, . . . , s✳

◆♦ss❛ ❛t❡♥çã♦ é ❞✐r✐❣✐❞❛ ❛♦ ❡st✉❞♦ ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s(f1, . . . , fs)✱ q✉❛♥❞♦ ❝❛❞❛

fi é ✉♠❛ ❞♦❜r❛✱ i= 1, . . . , s✳

❉❡✜♥✐çã♦ ✶✳✶✸✿ ❙❡❥❛♠ ϕ1, . . . , ϕs ✐♥✈♦❧✉çõ❡s ❡♠ (Rn,0) ❡ (f1, . . . , fs) ✉♠ ❞✐❛❣r❛♠❛

❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s✳ ❉✐③❡♠♦s q✉❡ (f1, . . . , fs) é ❛ss♦❝✐❛❞♦ à s✲✉♣❧❛ (ϕ1, . . . , ϕs)✱ ♦✉

(ϕ1, . . . , ϕs) é ❛ss♦❝✐❛❞❛ ❛ (f1, . . . , fs) s❡ fi é ✉♠❛ ❞♦❜r❛ ❛ss♦❝✐❛❞❛ ❛ ϕi ♣❛r❛ t♦❞♦ i =

❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✼

❆❣♦r❛ ❛♣r❡s❡♥t❛♠♦s ♦ ❢✉♥❞❛♠❡♥t❛❧ r❡s✉❧t❛❞♦ q✉❡ ❡st❛❜❡❧❡❝❡ q✉❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ♣♦❞❡ s❡r r❡❞✉③✐❞♦ à ❝❧❛ss✐✜❝❛çã♦ ❞❡s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s às q✉❛✐s ❡stã♦ ❛ss♦❝✐❛❞♦s✳

❚❡♦r❡♠❛ ✶✳✶✹✿ ❙❡❥❛(f1, . . . , fs)✉♠ ❞✐❛❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞♦ ❛(ϕ1, . . . , ϕs)

❡(g1, . . . , gs)✉♠ ❞✐❛❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞♦ ❛(ψ1, . . . , ψs)✳ ❊♥tã♦✱(f1, . . . , fs)

❡ (g1, . . . , gs) sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡✱ ❡ só s❡✱ (ϕ1, . . . , ϕs) ❡ (ψ1, . . . , ψs) sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛(f1, . . . , fs)❡(g1, . . . , gs)❡q✉✐✈❛❧❡♥t❡s✳ ❊♥tã♦✱ ❡①✐st❡♠ ❣❡r♠❡s

❞❡ ❞✐❢❡♦♠♦r✜s♠♦s h, k1, . . . , ks ❞❡ (Rn,0)t❛✐s q✉❡

gi =ki◦fi◦h−1 ∀i= 1, . . . , s.

▲♦❣♦ ❛ s✲✉♣❧❛ ❞❡ ✐♥✈♦❧✉çõ❡s(h◦ϕ1◦h−1, . . . , h◦ϕs◦h−1)é ❛ss♦❝✐❛❞❛ ❛(g1. . . , gs)✳ P❡❧❛

✉♥✐❝✐❞❛❞❡ ♠❡♥❝✐♦♥❛❞❛ ♥❛ Pr♦♣♦s✐çã♦ ✶✳✻ s❡❣✉❡ q✉❡

ψi =h◦ϕi h−1 ∀i= 1, . . . , s,

❡ ❛ss✐♠ t❡♠♦s q✉❡ (ϕ1. . . , ϕs) ❡ (ψ1, . . . , ψs) sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛h❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡(Rn_,0)t❛❧ q✉❡_{ψi =h◦ϕi◦h}−1

♣❛r❛i= 1. . . , s✳ ❊♥tã♦✱ ♦ ❞✐❛❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s(f1◦h−1, . . . , fs◦h−1)é ❛ss♦❝✐❛❞♦

❛ (ψ1. . . , ψs)✳ P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✶✶✱

(g1. . . , gs)∈ L ·(f1◦h−1, . . . , fs◦h−1).

P♦rt❛♥t♦✱ (f1, . . . , fs)❡ (g1, . . . , gs)sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

❈♦♠♦ ❝♦♥s❡qüê♥❝✐❛ ❞❡st❡ r❡s✉❧t❛❞♦✱ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s (f1. . . , fs)t❡♠♦s q✉❡✱ ♦ tr❛ç♦tr(d(ϕ1◦. . .◦ϕs)(0))é ✉♠ ✐♥✈❛r✐❛♥t❡ ♣♦r ❡q✉✐✈❛❧ê♥❝✐❛✱ ♦♥❞❡

(ϕ1, . . . , ϕs) é ❛ s✲✉♣❧❛ ❛ss♦❝✐❛❞❛ ❝♦♠ (f1, . . . , fs)✳

✶✳✸ ❈♦♥❥✉♥t♦s ❞❡ ✐♥✈♦❧✉çõ❡s tr❛♥s✈❡rs❛✐s

✽ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

❉❡✜♥✐çã♦ ✶✳✶✺✿ ❯♠ ❝♦♥❥✉♥t♦ Gs = {ϕ1, . . . , ϕs} ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ (Rn,0)✱ s ≤ n✱

é tr❛♥s✈❡rs❛❧ s❡ F(ϕi) é tr❛♥s✈❡rs❛❧ ❛ F(ϕj) ❡♠ 0 ♣❛r❛ i 6= j ❡ codim∩si=1T0F(ϕi) =

Ps

i=1codim(F(ϕi))✱ ♦♥❞❡ T0F(ϕi)❞❡♥♦t❛ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❞❡ F(ϕi) ❡♠ 0✳

◆♦t❡♠♦s q✉❡ ♣❛r❛ s = 2✱ ✐st♦ é✱ ♣❛r❛ ❞✉❛s ✐♥✈♦❧✉çõ❡s ϕ1, ϕ2 : (Rn,0) → (Rn,0)✱ ❛

tr❛♥s✈❡rs❛❧✐❞❛❞❡ s❡ r❡❞✉③ ❛

Rn _{=T0F(ϕ1) +T0F(ϕ2).}

◆♦t❡♠♦s q✉❡ ♥♦ ❝❛s♦ ❞❡ s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s ♥ã♦ é ♥❡❝❡ssár✐♦ ♦ tr❛t❛♠❡♥t♦ ❞❡ ❣❡r♠❡ ♣❛r❛ ♥♦ss❛ ❝❧❛ss✐✜❝❛çã♦✳

✶✳✸✳✶ ❋♦r♠❛s ♥♦r♠❛✐s ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s tr❛♥s✈❡rs❛✐s ❝♦♠ ❞✐✲

♠❡♥sã♦ ❞♦ s✉❜❡s♣❛ç♦ ❞❡ ♣♦♥t♦s ✜①♦s ❛r❜✐trár✐❛

❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♣❛r❛ ❝✉❥❛ ♣r♦✈❛ ❞❡♥♦t❛♠♦s ♣♦r V0 _{⊂ V}∗ _{♦ ❝♦♥✲} ❥✉♥t♦ ❞♦s ❢✉♥❝✐♦♥❛✐s ❧✐♥❡❛r❡s q✉❡ s❡ ❛♥✉❧❛♠ s♦❜r❡ ♦ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ V ❞❡ Rn✱ ✐st♦

é✱

V0 ={f;f(p) = 0, p∈V}.

❉❛ á❧❣❡❜r❛ ❧✐♥❡❛r t❡♠♦s q✉❡ dimV0 = codimV ✭❱❡❥❛ ❬✾✱ ♣✳ ✶✵✶❪✮✳

Pr♦♣♦s✐çã♦ ✶✳✶✻✿ ❙❡❥❛♠ϕ1, ϕ2 ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s tr❛♥s✈❡rs❛✐s ❡♠Rnt❛✐s q✉❡dimF(ϕ1)✲

= r ❡ dimF(ϕ2) = s✳ ❊♥tã♦✱ (ϕ1, ϕ2) é ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣❛r (ψ1, ψ2) t❛❧ q✉❡

F(ψ1) ❡ F(ψ2) sã♦ ❞❛❞♦s ♣♦r x1 = . . . xn−r = 0 ❡ xn−r+1 = . . . x2n−r−s = 0✱ r❡s♣❡❝t✐✈❛✲

♠❡♥t❡✳ P♦rt❛♥t♦✱ ψ1 ❡ ψ2 tê♠ ♠❛tr✐③❡s ❞♦ t✐♣♦

ψ1 =

  

−In−r 0 0

A2 In−s 0

A3 0 Ir+s−n

 

, ψ2 =

  

In−r B1 0

0 −In−s 0

0 B3 Ir+s−n

 

. ✭✶✳✷✮

❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ ♣r♦✈❛♠♦s q✉❡ (ϕ1, ϕ2) é ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣❛r

(ψ1, ψ2)t❛❧ q✉❡F(ψ1)❡F(ψ2)sã♦ ❞❛❞♦s ♣♦rx1 =. . . xn−r = 0❡xn−r+1 =. . . x2n−r−s= 0

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡s❞❡ q✉❡dimF(ϕ1) =r❡dimF(ϕ2) = s✱ ♣♦r ✉♠ r❡s✉❧t❛❞♦ ❞❛ á❧❣❡❜r❛

❧✐♥❡❛r ✭❬✾✱ ♣✳ ✶✵✾❪✮✱ ♦s s✉❜❡s♣❛ç♦s ❛♥✉❧❛❞♦r❡s ❞❡ F(ϕ1) ❡ F(ϕ2) sã♦ t❛✐s q✉❡

codimF(ϕ1)0 =n−r ❡ codimF(ϕ2)0 =n−s

❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✾

q✉❡

F(ϕ1) = {p∈Rn;f1(p) = · · ·=fn−r(p) = 0},

F(ϕ2) = {p∈Rn;fn−r+1(p) =· · ·=f2n−r−s(p) = 0}.

❉❡s❞❡ q✉❡ ϕ1 ❡ ϕ2 sã♦ tr❛♥✈❡rs❛✐s✱

codimF(ϕ1)∩ F(ϕ2) = codimF(ϕ1) + codimF(ϕ2) = 2n−r−s≤n.

❆✜r♠❛çã♦✿ {f1, . . . , f2n−r−s}é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❡♠(Rn)∗✳ ❉❡ ❢❛t♦✱ ♣r✐♠❡✐r♦

♥♦t❡♠♦s q✉❡

F(ϕ1)∩ F(ϕ2) = {p∈Rn;fi(p) = 0, 1≤i≤2n−r−s}.

❆❧é♠ ❞✐ss♦✱ codimF(ϕ1) ∩ F(ϕ2) = dim(F(ϕ1) ∩ F(ϕ2))∗ = 2n − r − s✱ ♣♦rt❛♥t♦✱

f1, . . . , f2n−r−ssã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❆❣♦r❛ ♣❡❣❛♠♦s ❛s ❢✉♥çõ❡sf2n−r−s+1, . . . , fn

❞❡ ♠♦❞♦ q✉❡ f1, . . . , fn ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ❞❡(Rn)∗✱ ♦♥❞❡ ♣❛r❛ ❡st❛ ❜❛s❡ ❡①✐st❡ ✉♠❛ ❜❛s❡

{v1, . . . , vn} ❞❡ Rn✳ ❈♦♥s✐❞❡r❡ ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ {π1, . . . , πn} ❞❡ (Rn)∗✱ ♦♥❞❡ πi ❞❡♥♦t❛ ❛

♣r♦❥❡çã♦ ❝❛♥ô♥✐❝❛ πi(x1, . . . , xn) = xi✱ ♣❛r❛ i = 1, . . . , n✳ ❆❣♦r❛✱ s❡❥❛ h−1 ❛ ♠❛tr✐③ ❞❡

♠✉❞❛♥ç❛ ❞❡ ❜❛s❡ ❞❡ {e1, . . . , en} ♣❛r❛ {v1, . . . , vn}✱ ❛ q✉❛❧ ✐♥❞✉③ ♦ ♦♣❡r❛❞♦r ✐♥✈❡rtí✈❡❧

(h−1₎∗ _{q✉❡ ❧❡✈❛ ❡❧❡♠❡♥t♦s ❞❛ ❜❛s❡ {f}

1, . . . , fn} ♥❛ ❜❛s❡ {π1, . . . , πn} ❞❡ (Rn)∗✱ q✉❡ ❡st❛

❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

(h−1)∗fi =fi◦h−1 =πi, i= 1, . . . , n.

❚♦♠❛♥❞♦ψi =h◦ϕi◦h−1 ♣❛r❛ i= 1,2✱(ϕ1, ϕ2)é ❡q✉✐✈❛❧❡♥t❡ ❛ (ψ1, ψ2)✳ ❆❣♦r❛ ❢❛❧t❛

♣r♦✈❛r q✉❡

F(ψ1) = {p∈Rn; x1 =. . .=xn−r = 0} ❡

F(ψ2) ={p∈Rn; xn−r+1 =. . .=x2n−r−s = 0}.

❙❡ ♣❡❣❛♠♦s v ∈ F(ψ1) = h(F(ϕ1))✱ t❡♠♦s q✉❡ v = h(u) ❝♦♠ u ∈ F(ϕ1)✳ ❙❡❥❛ fi ∈

(F(ϕ1))0✳ ❱❡❥❛♠♦s q✉❡

πi(v) = fi◦h−1(v) = fi(u) = 0,

♦ q✉❛❧ ♣r♦✈❛ q✉❡

F(ψ1)⊆ {p∈Rn;x1 =. . .=xn−r = 0}.

❆❣♦r❛✱ s❡ v ∈ {p∈Rn_{; x1 =. . .=xn−r = 0}}✱ ❡♥tã♦ _{0 =πi(v) =fi◦h}−1_(v) ♣❛r❛ _1≤i≤

n−r✳ ❆ss✐♠✱ v ∈ h(F(ϕ1)) =F(ψ1)✳ ❉❡ ❢♦r♠❛ s✐♠✐❧❛r s❡ ♣r♦✈❛ ❛ s❡❣✉♥❞❛ ✐❣✉❛❧❞❛❞❡✱ ♦

✶✵ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

❙❛❜❡♠♦s q✉❡ ψ1(0, . . . ,0, xn−r+1, . . . , xn) = (0, . . . ,0, xn−r+1, . . . , xn)✳ ❙❡ ❡s❝r❡✈❡♠♦s

ψ1 ❡♠ ❜❧♦❝♦s

ψ1 =

  

A1 A4 A7

A2 A5 A8

A3 A6 A9

  ,

♦♥❞❡ A1 ∈M(n−r)✱A5 ∈M(n−s) ❡A9 ∈M(r+s−n)✱ ❡♥tã♦

ψ1(0, . . . ,0, xn−r+1, . . . , x2n−r−s, . . . , xn) = (A4(xn−r+1, . . . , x2n−r−s) +A7(x2n−r−s+1, . . . , xn),

A5(xn−r+1, . . . , x2n−r−s) +A8(x2n−r−s+1, . . . , xn),

A6(xn−r+1, . . . , x2n−r−s) +A9(x2n−r−s+1, . . . , xn))

= (0, . . . ,0, xn−r+1, . . . , x2n−r−s, . . . , xn).

❆ss✐♠✱

A4(xn−r+1, . . . , x2n−r−s) +A7(x2n−r−s+1, . . . , xn) = (0, . . . ,0),

(A5−In−s)(xn−r+1, . . . , x2n−r−s) +A8(x2n−r−s+1, . . . , xn) = (0, . . . ,0),

A6(xn−r+1, . . . , x2n−r−s) + (A9−Ir+s−n)(x2n−r−s+1, . . . , xn) = (0, . . . ,0).

❉❡s❞❡ q✉❡ xn−r+1, . . . , x2n−r−s, x2n−r−s+1, . . . , xn sã♦ ❛r❜✐trár✐♦s✱ t❡♠♦s q✉❡

A4 = 0, A7 = 0;

A5 = In−s, A8 = 0;

A6 = 0, A9 = Ir+s−n.

P♦rt❛♥t♦✱

ψ1 =

  

A1 0 0

A2 In−s 0

A3 0 Ir+s−n

  .

❆♥❛❧♦❣❛♠❡♥t❡ ❢❛③❡♠♦s ❛s ❝♦♥t❛s ♣❛r❛ ψ2✱ ❧♦❣♦ψ1 ❡ψ2 tê♠ ♠❛tr✐③❡s ❞♦s t✐♣♦s

ψ1 =

  

A1 0 0

A2 In−s 0

A3 0 Ir+s−n

 

, ψ2 =

  

In−r B1 0

0 B2 0

0 B3 Ir+s−n

  .

❙❡♥❞♦ ψ1 ❡ ψ2 ✐♥✈♦❧✉çõ❡s✱ t❡♠♦s q✉❡

(a)A2₁ = In−r (b)A2+A2A1 = 0, A3+A3A1 = 0

❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✶

❖❜s❡r✈❛♥❞♦ ♦ ✐t❡♠ ✭a✮ ✈❡♠♦s q✉❡ ♣❛r❛ ♣r♦✈❛r q✉❡ A1 =−In−r é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡

F(A1) ={0}✳ ❉❡ ❢❛t♦✱ s❡ (x1, . . . , xn−r)∈ F(A1)✱ t❡♠♦s

ψ1(x1, . . . , xn−r,0. . . ,0) = (A1(x1, . . . , xn−r), A2(x1, . . . , xn−r), A3(x1, . . . , xn−r))

= (x1, . . . , xn−r, A2(x1, . . . , xn−r), A3(x1, . . . , xn−r)).

❉♦ ✐t❡♠ ✭b✮ t❡♠♦s q✉❡

(A2+A2A1)(x1, . . . , xn−r) = 0⇒A2(x1, . . . , xn−r) = 0,

(A3+A3A1)(x1, . . . , xn−r) = 0⇒A3(x1, . . . , xn−r) = 0.

❆ss✐♠✱(x1, . . . , xn−r,0, . . . ,0)∈ F(ψ1)✱ ♦ q✉❛❧ ✐♠♣❧✐❝❛ q✉❡x1 =. . .=xn−r = 0✳ P♦rt❛♥t♦✱

F(A1) ={0}✳

❆♥❛❧♦❣❛♠❡♥t❡✱ ✉s❛♥❞♦ ✭c✮ ❡ ✭d✮ ♦❜t❡♠♦s q✉❡B2 =−In−s✳

❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛ ❝♦♥❞✐çã♦ ♥❡❝❡ss❛r✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ❞♦✐s ♣❛r❡s ❞❡ ✐♥✈♦❧✉✲ çõ❡s ❧✐♥❡❛r❡s ❝♦♠ ❝♦❞✐♠❡♥sã♦ ❞♦ ❝♦♥❥✉♥t♦s ❞♦s ♣♦♥t♦s ✜①♦ ❛r❜✐trár✐❛ s❡❥❛♠ ❡q✉✐✈❛❧❡♥t❡s✿

Pr♦♣♦s✐çã♦ ✶✳✶✼✿ ❙❡❥❛♠ (ψ1, ψ2) ❡ (ψ1′, ψ′2) ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s ❡♠ Rn ❝♦♠

♠❛tr✐③❡s ❝♦♠♦ ❡♠ ✭✶✳✷✮✿

ψ1 =

  

−In−r 0 0

A2 In−s 0

A3 0 Ir+s−n

 

, ψ2 =

  

In−r B1 0

0 −In−s 0

0 B3 Ir+s−n

  

❡

ψ₁′ =

  

−In−r 0 0

A′

2 In−s 0

A′

3 0 Ir+s−n

 

, ψ₂′ =

  

In−r B′1 0

0 −In−s 0

0 B′

3 Ir+s−n

  .

❊♥tã♦ (ψ1, ψ2) ❡ (ψ1′, ψ′2) sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡✱ ❡ só s❡✱ ❡①✐st❡ H ♠❛tr✐③ ✐♥✈❡rtí✈❡❧

H =

  

(α1)n−r 0

0 (α2)n−s 0

δ γ βr+s−n

  .

t❛❧ q✉❡

A′₂α1 =α2A2,

B₁′α2 =α1B1,

A′₃α1 =−2δ+γA2+βA3,

✶✷ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

❉❡♠♦♥str❛çã♦✿ ❆ ♣r♦✈❛ é ❝♦♥s❡qüê♥❝✐❛ ❞✐r❡t❛ ❞❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ♣❛r❡s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✳

❆❣♦r❛ ✈♦❧t❛♠♦s ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ❝♦♠ ❝♦❞✐♠❡♥sã♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✜①♦s ✐❣✉❛❧ ❛ ✶✱ q✉❡ sã♦ ❛q✉❡❧❛s q✉❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s às ❞♦❜r❛s✳

✶✳✸✳✷ ■♥✈♦❧✉çõ❡s

ϕ

❝♦♠

codim

F

(

ϕ

) = 1

Pr♦♣♦s✐çã♦ ✶✳✶✽✿ ❙❡❥❛ Gs = {ϕ1, . . . , ϕs} ✉♠ ❝♦♥❥✉♥t♦ tr❛♥s✈❡rs❛❧ ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐✲

♥❡❛r❡s ❡♠ Rn✳ ❊♥tã♦✱ _{(ϕ1, . . . , ϕs)} é ❧✐♥❡❛r♠❡♥t❡ ❡q✉✐✈❛❧❡♥t❡ ❛ _{(ψ1, . . . , ψs)} t❛❧ q✉❡✱ ♣❛r❛

❝❛❞❛ ψi✱ F(ψi) é ❞❛❞♦ ♣❡❧❛ ❡q✉❛çã♦ xi = 0 ❡✱ ♣♦rt❛♥t♦✱ ψi t❡♠ ❛ ❢♦r♠❛

ψi(x1, . . . , xn) = (x1+ai1xi, . . . ,−xi, . . . , xn+ainxi), ✭✶✳✸✮

♣❛r❛ ❝♦♥st❛♥t❡s aij✱ j 6=i✱ 1≤j ≤n✳

❉❡♠♦♥str❛çã♦✿ ❆ ♣r♦✈❛ é s✐♠✐❧❛r á Pr♦♣♦s✐çã♦ ✶✳✶✻✳

❉❡♣♦✐s ❞❡ ♦❜t❡r ❛s ❢♦r♠❛s ♣r❡♥♦r♠❛✐s ❞❡ s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s tr❛♥s✈❡rs❛✐s✱ ♣r♦ss❡❣✉✐♠♦s ❛ ❛♣r❡s❡♥t❛r ❛s ❢♦r♠❛s ♣r❡♥♦r♠❛✐s ❞♦s ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s ❛ss♦❝✐❛❞❛s ❛ ❡st❛s s✲✉♣❧❛s✳

Pr♦♣♦s✐çã♦ ✶✳✶✾✿ ❙❡❥❛ Gs = {ϕ1, . . . , ϕs} ❝♦♠♦ ♥❛ Pr♦♣♦s✐çã♦ ✶✳✶✽✳ ❊♥tã♦ q✉❛❧q✉❡r

❞✐❛❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s (f1, . . . , fs) ❛ss♦❝✐❛❞♦ ❛ (ϕ1, . . . , ϕs) é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❞✐❛✲

❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s (g1, . . . , gs)❛ss♦❝✐❛❞♦ ❛ (ψ1, . . . , ψs)✱ ♦♥❞❡ψi é ❞❛❞♦ ♣♦r ✭✶✳✸✮

❡

gi(x1, . . . , xn) = (x1+

ai1

2 xi, . . . , x

2

i, . . . , xn+

ain

2 xi). ✭✶✳✹✮

❉❡♠♦♥str❛çã♦✿ P❛r❛ ❝❛❞❛ i = 1, . . . , s✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ✐s♦♠♦r✜s♠♦ ❧✐♥❡❛r hi ❞❡ Rn

❞❛❞♦ ♣♦r

hi(x1, . . . , xn) = (x1−

ai1

2 xi, . . . , xi, . . . , xn− ain

2 xi), ♣❛r❛ ♦ q✉❛❧

h−_i 1(x1, . . . , xn) = (x1+

ai1

2 xi, . . . , xi, . . . , xn+ ain

2 xi). ❊♥tã♦✱ ♦❜t❡♠♦s ψi ❝♦♠♦ ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✿

ψi =hi◦ϕ0i ◦h−1(x1, . . . , xn) = hi◦ϕ0i(x1+

ai1

2 xi, . . . , xi, . . . , xn+ ain

2 xi) =hi(x1+

ai1

2 xi, . . . ,−xi, . . . , xn ain

❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✸

▲♦❣♦ (ϕ0

1, . . . , ϕ0s) é ❡q✉✐✈❛❧❡♥t❡ ❛ (ψ1, . . . , ψs) ♣♦❡ h✳ ❆ss✐♠✱ ❛ ❞♦❜r❛ gi ❞❡✜♥✐❞❛ ♣♦r

gi =fi0◦h−i1 é ❛ss♦❝✐❛❞❛ ❛ ψi✱ ♣♦✐s gi◦ψi = fi0 ◦h−i 1◦hi◦ϕ0i ◦h−i 1 =gi ❡✱ ♣♦rt❛♥t♦✱ ♦

❞✐❛❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s(g1, . . . , gs)é ❛ss♦❝✐❛❞♦ às✲✉♣❧❛ ❞❡ ✐♥✈♦❧✉çõ❡s(ψ1, . . . , ψs)✳

▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✹ ♦s ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s (g1, . . . , gs) ❡ (f1, . . . , fs)

sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

❖s s❡❣✉✐♥t❡s ❞♦✐s r❡s✉❧t❛❞♦s ❞ã♦ ❛s ❢♦r♠❛s ♥♦r♠❛✐s ❞❡s✲✉♣❧❛s ❞❡ ✐♥✈♦❧✉çõ❡s ❡ ❞♦s ❝♦✲ rr❡s♣♦♥❞❡♥t❡s ❞✐❛❣r❛♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ❞♦❜r❛s q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✐♥✈♦❧✉çõ❡s é tr❛♥s✲ ✈❡rs❛❧ ❡ ❣❡r❛ ✉♠ ❣r✉♣♦ ❆❜❡❧✐❛♥♦✱ ❛s ✐♥✈♦❧✉çõ❡s ♥ã♦ s❡♥❞♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❧✐♥❡❛r❡s✳

❚❡♦r❡♠❛ ✶✳✷✵✿ ❙❡ Gs = {ϕ1, . . . , ϕs} é tr❛♥s✈❡rs❛❧ ❡ ∆s = hϕ1, . . . , ϕsi é ❆❜❡❧✐❛♥♦✱

❡♥tã♦ (ϕ1, . . . , ϕs) é ❡q✉✐✈❛❧❡♥t❡ ❛ (ϕ01, . . . , ϕ0s)✱ ♦♥❞❡

ϕ0_i(x1, . . . , xn) = (x1, . . . ,−xi, . . . , xn), i= 1, . . . , s. ✭✶✳✺✮

❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✱Gs ={ϕ1, . . . , ϕs}é ❡q✉✐✈❛❧❡♥t❡ ❛G¯s ={ϕ¯1, . . . ,ϕ¯s}✱

❝♦♠ϕ¯i✬s s❡♥❞♦ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s✳ ❈♦♠♦Gsé tr❛♥s✈❡rs❛❧✱ ❛✜r♠❛♠♦s q✉❡G¯sé tr❛s✈❡rs❛❧✳

❉❡ ❢❛t♦✱ s❡❥❛ h♦ ❣❡r♠❡ ❞❡ ❞✐❢❡♦♠♦r✜s♠♦ t❛❧ q✉❡ ϕi =h◦ϕ¯i◦h−1✳ ❚❡♠♦s

h:F( ¯ϕi)→ F(ϕi),

❡ é ❢á❝✐❧ ✈❡r q✉❡ dh(0)(T0F( ¯ϕi)) = dh(0)(F( ¯ϕi)) = T0F(ϕi)✱ ♣❛r❛ ❝❛❞❛ 1 ≤ i ≤ s✳ ❊

t❛♠❜é♠ ✈❡❥❛♠♦s q✉❡ ♣❡❧❛ tr❛♥✈❡rs❛❧✐❞❛❞❡ ❞❡ Gs ❡ ♣❡❧❛ ❧✐♥❡❛r✐❞❛❞❡ ❞❡ϕ¯i✿

codim

s

\

i=1

T0F( ¯ϕi) = codim s

\

i=1

F( ¯ϕi) = codimdh(0)( s

\

i=1

F( ¯ϕi))

= codim

s

\

i=1

dh(0)(F( ¯ϕi)) = codim s

\

i=1

T0F(ϕi)

=

s

X

i=1

codimF(ϕi) = s

X

i=1

codimF( ¯ϕi).

❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ ♣❛r❛ i6=j

Rn_{=T0F(ϕi) +T0F(ϕj) =dh(0)(T0F( ¯ϕi)) +dh(0)(T0F( ¯ϕj))}

=dh(0)(T0F( ¯ϕi) +T0F( ¯ϕj)) = T0F( ¯ϕi) +T0F( ¯ϕi);

♣♦rt❛♥t♦✱ _G¯_{s é tr❛♥s✈❡rs❛❧✳ ▼❛✐s ❛✐♥❞❛✱∆}¯_{s é ❆❜❡❧✐❛♥♦✱ ♣♦✐s}

¯

ϕi◦ϕ¯j =h−1◦ϕi◦h◦h−1◦ϕj ◦h=h−1◦ϕj◦ϕi◦h= ¯ϕj ◦ϕ¯i.

✶✹ ❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s

❝♦♠♦ ❡♥ ✭✶✳✸✮ ❡ ✉s❛r ♦ ❢❛t♦ ❞❡ q✉❡ _G¯_{s é ❆❜❡❧✐❛♥♦ ♣❛r❛ ❝♦♥s❡❣✉✐r q✉❡ aij = 0✳ ❆ ♦✉tr❛} é ♦❜s❡r✈❛r q✉❡ t♦❞♦s ♦s ϕ¯i sã♦ ❞✐❛❣♦♥❛❧✐③á✈❡✐s ❝♦♠ n−1 ❛✉t♦✈❛❧♦r❡s ✶ ❡ ✉♠ ❛✉t♦✈❛❧♦r

−1✳ ❯s❛♥❞♦ ❞✐❛❣♦♥❛❧✐③❛çã♦ s✐♠✉❧tâ♥❡❛ ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❛❣♦♥✐③á✈❡✐s ❡ ✉♠❛ ♠❛tr✐③ ❞❡ ♣❡r♠✉t❛çã♦ s❡ ♥❡❝❡ssár✐♦✱ ♦❜t❡♠♦s q✉❡ ( ¯ϕ1, . . . ,ϕ¯s) é ❡q✉✐✈❛❧❡♥t❡ ❛(ϕ01, . . . , ϕ0s)✳

P❡❧♦s t❡♦r❡♠❛s ✶✳✶✹ ❡ ✶✳✷✵ t❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿

❚❡♦r❡♠❛ ✶✳✷✶✿ ❙❡ Gs é tr❛♥s✈❡rs❛❧ ❡ ∆s =hϕ1, . . . , ϕni é ❆❜❡❧✐❛♥♦✱ ❡♥tã♦ q✉❛❧q✉❡r ❞✐❛✲

❣r❛♠❛ ❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s(f1, . . . , fs)❛ss♦❝✐❛❞♦ ❛(ϕ1, . . . , ϕs)é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❞✐❛❣r❛♠❛

❞✐✈❡r❣❡♥t❡ ❞❡ ❞♦❜r❛s (f0

1, . . . , fs0)✱ ♦♥❞❡