❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊
❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼
▼❆❚❊▼➪❚■❈❆
❋■❇❘❆❉❖❙✱ ❈▲❆❙❙❊❙ ❉❊
❙❚■❊❋❊▲✲❲❍■❚◆❊❨ ❊ ❘❊❙❯▲❚❆❉❖❙ ❉❊
◆➹❖ ■▼❊❘❙➹❖
❈❆■❖ ❈❆❘▲❊❱❆❘❖ ■◆❋❖❘❩❆❚❖
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊
❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼
▼❆❚❊▼➪❚■❈❆
❋■❇❘❆❉❖❙✱ ❈▲❆❙❙❊❙ ❉❊
❙❚■❊❋❊▲✲❲❍■❚◆❊❨ ❊ ❘❊❙❯▲❚❆❉❖❙ ❉❊
◆➹❖ ■▼❊❘❙➹❖
❈❆■❖ ❈❆❘▲❊❱❆❘❖ ■◆❋❖❘❩❆❚❖
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ PP●▼ ❞❛ ❯❋❙❈❛r ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛çã♦✿ Pr♦❢❛✳ ❉r❛✳ ❆❞r✐❛♥❛ ❘❛♠♦s✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar
I43fc
Inforzato, Caio Carlevaro.
Fibrados, classes de Stiefel-Whitney e resultados de não imersão / Caio Carlevaro Inforzato. -- São Carlos : UFSCar, 2012.
63 f.
Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2012.
1. Topologia. 2. Fibrados vetoriais. 3. Classes de Stiefel-Whitney. 4. Variedades diferenciáveis. I. Título.
❆❣r❛❞❡❝✐♠❡♥t♦s
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛❀
➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ Pr♦❢❛✳ ❉r❛✳ ❆❞r✐❛♥❛ ❘❛♠♦s✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ♣❡❧❛ ❣r❛♥❞❡ ❛❥✉❞❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦❀
❆♦ ♣r♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❡ s❡✉s ♣r♦❢❡ss♦r❡s✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ r❡❛❧✐③❛r ❡st❡ tr❛❜❛❧❤♦❀
❘❡s✉♠♦
❆♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ ✐♥tr♦❞✉tór✐♦ ❞❡ ❱❛r✐❡❞❛❞❡s ❙✉❛✈❡s✱ ❋✐❜r❛❞♦s ❡ ❈❧❛s✲ s❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ✭❞❡ ✜❜r❛❞♦s ✈❡t♦r✐❛s r❡❛✐s✮✳
❊①♣❧✐❝❛♠♦s q✉❡✱ ❞❛❞❛ ✉♠❛ ❝❡rt❛ ✈❛r✐❡❞❛❞❡ s✉❛✈❡ m✲❞✐♠❡♥s✐♦♥❛❧✱ ❛s ❝❧❛ss❡s
❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ❞♦ s❡✉ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ♣♦❞❡♠ s❡r ✉s❛❞❛s ♣❛r❛ ❣❛r❛♥t✐r q✉❡ t❛❧ ✈❛r✐❡❞❛❞❡ ♥ã♦ ✐♠❡r❣❡ ✭s✉❛✈❡♠❡♥t❡✮ ❡♠ ❝❡rt♦s ❡s♣❛ç♦s ❊✉❝❧✐❞✐❛♥♦sRj✳ ◆❡ss❡ s❡♥t✐❞♦✱
❝♦♥s✐❞❡r❛♠♦s ❛ ✈❛r✐❡❞❛❞❡ ●r❛ss♠❛♥♥✐❛♥❛G2,n✱ ✈❛r✐❡❞❛❞❡ ❞♦s ✷✲s✉❜❡s♣❛ç♦s ❞❡Rn+2✱
❡ r❡❛❧✐③❛♠♦s ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❞❡ ♥ã♦ ✐♠❡rsã♦✱ ♣r♦✈❛❞♦ ♣♦r ❱✳ ❖♣r♦✐✉ ❬Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❊❞✐♥❜✉r❣❤ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✾✼✼❪✿
✧❙❡❥❛ n >1✉♠ ♥❛t✉r❛❧ ❡ ❝♦♥s✐❞❡r❡ s= 2r t❛❧ q✉❡ s ≤2n <2s✳ ❙❡ n6=s−1✱
❆❜str❛❝t
❲❡ ♣r❡s❡♥t ❛♥ ✐♥tr♦❞✉❝t♦r② st✉❞② ♦❢ s♠♦♦t❤ ♠❛♥✐❢♦❧❞s✱ ❜✉♥❞❧❡s ❛♥❞ ❙t✐❡❢❡❧✲ ❲❤✐t♥❡② ❝❧❛ss❡s ✭♦❢ r❡❛❧ ✈❡❝t♦r ❜✉♥❞❧❡s✮✳
❲❡ ❡①♣❧❛✐♥❡❞ t❤❛t✱ ❣✐✈❡♥ ❛ ❝❡rt❛✐♥ s♠♦♦t❤ ♠✲❞✐♠❡♥s✐♦♥❛❧ ♠❛♥✐❢♦❧❞✱ t❤❡ ❙t✐❡❢❡❧✲ ❲❤✐t♥❡② ❝❧❛ss❡s ♦❢ ✐ts t❛♥❣❡♥t ❜✉♥❞❧❡ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡♥s✉r❡ t❤❛t s✉❝❤ ❛ ♠❛♥✐❢♦❧❞ ❞♦❡s ♥♦t ✐♠♠❡rs❡ ✭s♠♦♦t❤❧②✮ ✐♥ ❝❡rt❛✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡s Rj✳ ■♥ t❤✐s s❡♥s❡✱ ✇❡
❝♦♥s✐❞❡r t❤❡ ●r❛ss♠❛♥♥ ♠❛♥✐❢♦❧❞ G2,n ♦❢ t❤❡ ✷✲s✉❜s♣❛❝❡s ♦❢ Rn+2✱ ❛♥❞ ✇❡ ❝❛rr②
♦✉t ❛ ❞❡t❛✐❧❡❞ st✉❞② ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦♥✲✐♠♠❡rs✐♦♥ t❤❡♦r❡♠✱ ♣r♦✈❡❞ ❜② ❱✳ ❖♣r♦✐✉ ❬Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❊❞✐♥❜✉r❣❤ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✾✼✼❪✿
✧▲❡t n > 1 ❜❡ ❛ ♥❛t✉r❛❧ ♥✉♠❜❡r ❛♥❞ ❝♦♥s✐❞❡r s = 2r s✉❝❤ t❤❛t s ≤ 2n < 2s✳
■❢n =s−1✱ t❤❡♥ G2,n ❞♦❡s ♥♦t ✐♠♠❡rs❡ ✐♥ R2s−3❀ ✐❢ n =s−1✱ t❤❡♥G2,n ❞♦❡s ♥♦t
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✽
✶ Pr❡❧✐♠✐♥❛r❡s ✶✵
✶✳✶ ❱❛r✐❡❞❛❞❡s ❙✉❛✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✶ ❋✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ P❧❛♥♦ ❚❛♥❣❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✷ ❆s ✈❛r✐❡❞❛❞❡s ●r❛ss♠❛♥♥✐❛♥❛sGk,n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✶✳✷ ❈♦❤♦♠♦❧♦❣✐❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ Z2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❋✐❜r❛❞♦s ✷✹
✷✳✶ ❋✐❜r❛❞♦ ❝♦♦r❞❡♥❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✶ ❈♦♥str✉çã♦ ❞❡ ✜❜r❛❞♦s ❛ ♣❛rt✐r ❞❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ✳ ✳ ✳ ✷✽ ✷✳✷ ❋✐❜r❛❞♦s ❱❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ❖ ✜❜r❛❞♦ ❝❛♥ô♥✐❝♦ s♦❜r❡ Gk,n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷✳✹ ❖ P✉❧❧✲❇❛❝❦ ❡ ❛ ❙♦♠❛ ❞❡ ❲❤✐t♥❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✺ ❖ ✜❜r❛❞♦ ♥♦r♠❛❧ ❞❡ ✉♠❛ ✐♠❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸ ❈❧❛ss❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ❡ r❡s✉❧t❛❞♦s ❞❡ ♥ã♦ ✐♠❡rsã♦ ✹✼
✸✳✶ ❈❧❛ss❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✷ ❘❡s✉❧t❛❞♦s ❞❡ ♥ã♦ ✐♠❡rsã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✸ ❙♦❜r❡ H∗
(Gk,n) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸✳✹ ❘❡s✉❧t❛❞♦s ❞❡ ♥ã♦ ✐♠❡rsã♦ ❞❡ G2,n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✸✳✺ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✹✳✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
■♥tr♦❞✉çã♦
◆❡st❛ ❞✐ss❡rt❛çã♦ r❡❛❧✐③❛♠♦s✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ✉♠ ❡st✉❞♦ ✐♥tr♦❞✉tór✐♦ s♦❜r❡ ❞✉❛s t❡♦r✐❛s ✧s♦✜st✐❝❛❞❛s✧❡ ✧♠♦❞❡r♥❛s✧✿ ✜❜r❛❞♦s ❡ ❝❧❛ss❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡②✳
❆❧é♠ ❞❡ s❡✉ ✈❛❧♦r ✐♥trí♥s❡❝♦✱ ❡ss❛s ❞✉❛s t❡♦r✐❛s ❥á ❡stã♦ ❝♦♥s♦❧✐❞❛❞❛s✱ ♣❡❧❛ ❧✐t❡✲ r❛t✉r❛✱ ❝♦♠♦ ❢❡rr❛♠❡♥t❛s út❡✐s ♥❛ ❛❜♦r❞❛❣❡♠ ❞❡ ♣r♦❜❧❡♠❛s ❞✐✈❡rs♦s ❡♠ ●❡♦♠❡tr✐❛ ❡ ❚♦♣♦❧♦❣✐❛✳
❊♠ ♣❛rt✐❝✉❧❛r✱ ❛❜♦r❞❛♠♦s✱ ♥❡st❡ tr❛❜❛❧❤♦✱ ❛ ❡str❛té❣✐❛ ❞❡ ✉t✐❧✐③❛r ❝❧❛ss❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ♥♦s ❝❤❛♠❛❞♦s ✧r❡s✉❧t❛❞♦s ❞❡ ♥ã♦ ✐♠❡rsã♦✧✳ ❚❛✐s r❡s✉❧t❛❞♦s ❡stã♦ ✐♥s❡r✐❞♦s ♥♦ ❝♦♥t❡①t♦ ❞♦ t❡♦r❡♠❛ ❝❧áss✐❝♦ ❞❡ ✐♠❡rsã♦ ❞❡ ❲❤✐t♥❡②✿
❚❡♦r❡♠❛✳ ❙❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ s✉❛✈❡ ❞❡ ❞✐♠❡♥sã♦ n > 1✱ ❡♥tã♦
❡①✐st❡ ✉♠❛ ✐♠❡rsã♦ φ :M −→R2n−1✳ ❬✶✷❪
❖❜s❡r✈❡ q✉❡✱ s❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡ s❡r ✐♠❡rs❛ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦Rm✱ ❡♥tã♦
❡❧❛ ♣♦❞❡ s❡r ✐♠❡rs❛ ❡♠Rj ♣❛r❛ t♦❞♦j > m✳
◆❡st❡ ♣♦♥t♦✱ ❝❛❜❡ ❛ ♣❡r❣✉♥t❛✿ ❞❛❞❛ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛Mn✱n✲❞✐♠❡♥s✐♦♥❛❧✱
❡s♣❡❝í✜❝❛✱ ❡❧❛ ♣♦❞❡ s❡r ✐♠❡rs❛ ❡♠ Rn+k ❝♦♠ k < n−1❄
❖ ❡♥✉♥❝✐❛❞♦ ❞❛ q✉❡stã♦ ❛❝✐♠❛ é ♣✉r❛♠❡♥t❡ ❣❡♦♠étr✐❝♦❀ ♥♦ ❡♥t❛♥t♦✱ ❝♦♠♦ ♣r❡✲ t❡♥❞❡♠♦s ❡①♣❧✐❝❛r ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ✉♠❛ r❡s♣♦st❛ ♥❡❣❛t✐✈❛ ♣♦❞❡ s❡r ❣❛r❛♥t✐❞❛ ❛♦ ✐♥✈❡st✐❣❛r♠♦s ❝❡rt♦ ❡❧❡♠❡♥t♦ ❞♦ ❛♥❡❧ ❞❡ ❈♦❤♦♠♦❧♦❣✐❛ ✭❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ Z2✮
❞❛ ✈❛r✐❡❞❛❞❡Mn ❝♦♥s✐❞❡r❛❞❛❀ t❛❧ ❡❧❡♠❡♥t♦ ❡s♣❡❝✐❛❧ é ❛ ❝❧❛ss❡ ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ❞♦
✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❡Mn✳
❈♦♠♦ ❡①❡♠♣❧♦ ❝❛♥ô♥✐❝♦✱ ♥❛ ❙❡çã♦ ✸✳✷✱ ✉t✐❧✐③❛r❡♠♦s ❡ss❛ ❡str❛té❣✐❛ ♣❛r❛ ♠♦s✲ tr❛r q✉❡Mn=RP2r
✭❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ r❡❛❧ ❞❡ ❞✐♠❡♥sã♦n= 2r✮ ♥ã♦ ♣♦❞❡ s❡r ✐♠❡rs♦
❡♠Rn+k❝♦♠k < n−1❀ ♦✉ s❡❥❛✱ ♣❛r❛ ❡st❛ ✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛ ❡s♣❡❝í✜❝❛✱ ❛ ❝♦❞✐♠❡♥sã♦
❞❡ ✐♠❡rsã♦ ❣❛r❛♥t✐❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❲❤✐t♥❡② é ❛ ♠❡❧❤♦r ♣♦ssí✈❡❧✳
❙❡❣✉✐♥❞♦ ❡ss❛ ❧✐♥❤❛✱ ❖♣r♦✐✉ ♣r♦✈♦✉ ❡♠ ❬✶✹❪ r❡s✉❧t❛❞♦s ❞❡ ♥ã♦ ✐♠❡rsã♦ ♣❛r❛ ❛s ✈❛r✐❡❞❛❞❡s ●r❛ss♠❛♥♥✐❛♥❛s M2n = G
2,n ✭❡s♣❛ç♦ ❞♦s ✷✲s✉❜❡s♣❛ç♦s ❞❡ Rn+2✮✳ ▼❛✐s
♣r❡❝✐s❛♠❡♥t❡✱ ❡❧❡ ♣r♦✈♦✉ ♦ s❡❣✉✐♥t❡
❚❡♦r❡♠❛✳ ❙❡❥❛ n >1 ✉♠ ♥❛t✉r❛❧ ❡ ❝♦♥s✐❞❡r❡ s = 2r t❛❧ q✉❡ s≤2n < 2s✳ ❊♥tã♦✿
✶✳ G2,n ♥ã♦ ✐♠❡r❣❡ ❡♠ R2s−3✱ ♣❛r❛ n 6=s−1❀
✷✳ G2,s−1 ♥ã♦ ✐♠❡r❣❡ ❡♠ R3s−3✳
◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦✱ ♥❡st❡ tr❛❜❛❧❤♦✱ é ❞❡s❡♥✈♦❧✈❡r ♦s r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛♣r❡s❡♥t❛r ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❛❝✐♠❛✳
❖❜s❡r✈❡ q✉❡✿ ♣❛r❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛rn = 2r✱ ❡ss❡ t❡♦r❡♠❛ ❣❛r❛♥t❡ q✉❡G
2,n ♥ã♦
✐♠❡r❣❡ ❡♠R2n−3❀ ❞❛í✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ✐♠❡rsã♦ ❞❡ ❲❤✐t♥❡②✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ✐♠❡rsã♦
só ❝♦♥t✐♥✉❛ ✧❛❜❡rt♦✧♣❛r❛R2n−2✳
❆ s❡❣✉✐r✱ ✐♥❞✐❝❛♠♦s ♦ ♠♦❞♦ ❝♦♠ q✉❡ ❡st❛ ❞✐ss❡rt❛çã♦ ❡stá ♦r❣❛♥✐③❛❞❛✳
◆♦ ❝❛♣ít✉❧♦ ✶✱ Pr❡❧✐♠✐♥❛r❡s✱ r❡✉♥✐♠♦s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❞❡ ❈♦❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ✭❡ ❝❡❧✉❧❛r✮ ❜ás✐❝♦s ♣❛r❛ ♦ q✉❡ s❡❣✉❡❀ ❛❧é♠ ❞✐ss♦✱ ❞❡s❡♥✈♦❧✈❡♠♦s ✉♠ ❡st✉❞♦ ✐♥tr♦❞✉tór✐♦ s♦❜r❡ ✈❛r✐❡❞❛❞❡s s✉❛✈❡s✳
❖ ❝❛♣ít✉❧♦ ✷✱ ❋✐❜r❛❞♦s✱ é ❞❡❞✐❝❛❞♦ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ❋✐❜r❛❞♦ ❡♠ ✉♠ ❡♥❢♦q✉❡ ♠❛✐s ❛♠♣❧♦✱ ❝♦♥❢♦r♠❡ ❙t❡❡♥r♦♦❞ ❬✽❪✱ ❡✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛♦ ❝♦♥❝❡✐t♦ ❞❡ ❋✐❜r❛❞♦ ❱❡t♦r✐❛❧ ✭r❡❛❧✮✳
◆♦ t❡r❝❡✐r♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ ✐♥t✐t✉❧❛❞♦ ✧❈❧❛ss❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ❡ r❡✲ s✉❧t❛❞♦s ❞❡ ♥ã♦ ✐♠❡rsã♦✧✱ ❞❡✜♥✐♠♦s ❛①✐♦♠❛t✐❝❛♠❡♥t❡ ❝❧❛ss❡s ❞❡ ❙t✐❡❢❡❧✲❲❤✐t♥❡② ❡ ❞✐s❝♦rr❡♠♦s s♦❜r❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❡✈❛♥t❡s✳ ❆♣r❡s❡♥t❛♠♦s✱ ❡♠ s❡❣✉✐❞❛✱ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s s♦❜r❡ ♦ ❛♥❡❧ ❞❡ ❈♦❤♦♠♦❧♦❣✐❛ ❞❡ G2,n✱ ❝♦♥❢♦r♠❡ ❬✶✺❪ ✭❡♠ t❡r✲
♠♦s ❞♦s ❝♦❝✐❝❧♦s ❞❡ ❙❝❤✉❜❡rt✮✳ ❋✐♥❛❧♠❡♥t❡✱ ♥❛ ❙❡çã♦ ✸✳✹✱ r❡❛❧✐③❛♠♦s ♥♦ss♦ ♣r✐♥❝✐♣❛❧ ❡st✉❞♦✿ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❖♣r♦✐✉ ❡♥✉♥❝✐❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✳
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦ ❡st❛❜❡❧❡❝❡♠♦s ♥♦t❛çõ❡s✱ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❡❧❡♠❡♥t❛r❡s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❝❛♣ít✉❧♦s q✉❡ s❡❣✉❡♠✳ ❆❞♠✐t✐♠♦s q✉❡ ♦ ❧❡✐t♦r t❡♥❤❛ ❢❛✲ ♠✐❧✐❛r✐❞❛❞❡ ❝♦♠ tó♣✐❝♦s ❜ás✐❝♦s ❞❡ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ ❡ ❞❡ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ ✭❡♠ ❡s♣❡❝✐❛❧ ❝♦♠ ❍♦♠♦❧♦❣✐❛ ❡ ❈♦❤♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r❡s✮✳ ◆❡ss❡ ❝♦♥t❡①t♦✱ ✉s❛♠♦s ❡ ✐♥❞✐✲ ❝❛♠♦s ❬✼❪✱ ❬✹❪ ❡ ❬✻❪ ❝♦♠♦ ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s✳
✶✳✶ ❱❛r✐❡❞❛❞❡s ❙✉❛✈❡s
❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ X é ❞✐t♦ s❡r ✉♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛
❞❡ ❞✐♠❡♥sã♦ n s❡X é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ ❡ ❧♦❝❛❧♠❡♥t❡
♥✲❊✉❝❧✐❞✐❛♥♦✱ ✐st♦ é✱ ♣❛r❛ ❝❛❞❛ p∈X ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ U ❞❡ p ❤♦♠❡♦♠❛r❢❛
❛ ✉♠ ❛❜❡rt♦ ❞❡ Rn✳
▲❡♠❛ ✶✳✶✳✷✳ ❙❡❥❛ X ✉♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❞❡ ❞✐♠❡♥sã♦ n✳ ❊♥tã♦ X é ❧♦❝❛❧✲
♠❡♥t❡ ❝♦♥❡①♦✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦✱ r❡❣✉❧❛r ❡ ♠❡tr✐③á✈❡❧✳
❉❡♠✳✿ ❙❡❥❛♠ p∈ X✱ U ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p ❤♦♠❡♦♠♦r❢❛ ❛ U′ ✭❛❜❡rt♦ ❞❡ Rn✮ ❡
φ:U −→U′ ❤♦♠❡♦♠♦r✜s♠♦✳ ❊♥tã♦ ❞❛❞♦ V ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ pt❡♠♦s q✉❡ ❡①✐st❡
✉♠ ❛❜❡rt♦ W✱ ❝♦♠ W ⊂U∩V✱ ❡ φ(W) =Bǫ(φ(p)) ✭❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛ ❡♠ φ(p)
❞❡ r❛✐♦ǫ✮ ❝♦♠ φ(W) = Bǫ(φ(p))⊂U′✳
❆ss✐♠✱ W é ❝♦♥❡①♦ ❡ W é ❝♦♠♣❛❝t♦ ❡✱ ♣♦rt❛♥t♦✱ X é ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡
❧♦❝❛❧♠❡♥t❡ ❝♦♥❡①♦✳ ❈♦♠♦ X é ❞❡ ❍❛✉s❞♦r✛✱ s❡❣✉❡ q✉❡ t❛♠❜é♠ é r❡❣✉❧❛r✳ ❆ss✐♠✱
♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ♠❡tr✐③❛çã♦ ❞❡ ❯r②s♦❤♥✱ ❝♦♥❝❧✉í♠♦s q✉❡X é ♠❡tr✐③á✈❡❧✳
❙❛❜❡♠♦s q✉❡ t♦❞♦ ❡s♣❛ç♦ ♠étr✐❝♦ é ♣❛r❛❝♦♠♣❛❝t♦ ✭❚❡♦r❡♠❛ ❞❡ ❙t♦♥❡✮❀ ❛ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ❛♥t❡r✐♦r✱ t♦❞❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ é ✉♠ ❡s♣❛ç♦ ♣❛r❛❝♦♠♣❛❝t♦✳
❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ Rn é ✉♠❛ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ ❞❡
❞✐♠❡♥sã♦ n✳
❊①❡♠♣❧♦ ✶✳✶✳✹✳ ❙❡❥❛ Sn ⊂ Rn+1✱ ❛ ❡s❢❡r❛ ♥✲❞✐♠❡♥s✐♦♥❛❧✳ ❊♥tã♦ Sn é ✉♠❛
✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛ n✲❞✐♠❡♥s✐♦♥❛❧✳ ❉❡ ❢❛t♦✿ Sn é ✉♠ ❡s♣❛ç♦ ❞❡ ❍❛✉s❞♦r✛ ❝♦♠
❜❛s❡ ❡♥✉♠❡rá✈❡❧ ❡✱ ❛❧é♠ ❞✐ss♦✱ t♦❞♦ ♣♦♥t♦ p∈Sn ♣❡rt❡♥❝❡ ❛ ✉♠ ❞♦s ❝♦♥❥✉♥t♦s
Hi+ ={(x1, . . . , xn+1)∈Rn;xi >0}
Hi− ={(x1, . . . , xn+1)∈Rn;xi <0},
i∈1, . . . , n+ 1✳ ▼❛s ❝❛❞❛ ✉♠ ❞❡ss❡s ❝♦♥❥✉♥t♦s é ❤♦♠❡♦♠♦r❢♦ à ❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛
❡♠ 0 ❡ r❛✐♦1✱ B1(0) ❞❡ Rn ✭❜❛st❛ ❝♦♥s✐❞❡r❛r φ±i :H
±
i −→B1(0)✱ ❞❛❞❛ ♣♦rφ±i (x) =
(x1, . . . , xi−1, xi+1, . . . xn+1)✮✳ ▲♦❣♦ Sn é ❧♦❝❛❧♠❡♥t❡ n✲❊✉❝❧✐❞✐❛♥♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✺✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠ ❛t❧❛s C∞ ❞❡ ❞✐♠❡♥sã♦ n ∈ N
♣❛r❛ X é ✉♠❛ ❝♦❧❡çã♦ A = {(Uα, φα)} ❞❡ ❝❛rt❛s✱ ✐st♦ é✱ ♣❛r❡s ❞❛ ❢♦r♠❛ (Uα, φα)
❝♦♠ Uα ⊂ X ❛❜❡rt♦ ❞❡ X ❡ φα : Uα −→ φα(Uα) ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ ✉♠ ❛❜❡rt♦
φα(Uα) ❞❡ Rn✱ t❛❧ q✉❡✿
✶✳ ❆ ❝♦❧❡çã♦ {Uα} ❝♦❜r❡ X❀
✷✳ ❙❡Uα∩Uβ 6=∅❡♥tã♦φα◦φ−β1 :φβ(Uα∩Uβ)−→φα(Uα∩Uβ)é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦
❞❡ ❝❧❛ss❡ C∞ ✭✐st♦ é✱ ❛s ❝❛rt❛s sã♦
C∞✲❝♦♠♣❛tí✈❡✐s✮✳
❯♠ ❛t❧❛sAé ♠❛①✐♠❛❧ s❡ t♦❞❛ ❝❛rt❛(Uα, φα)❞❡X q✉❡ éC∞✲❝♦♠♣❛tí✈❡❧ ❝♦♠
q✉❛❧q✉❡r ❝❛rt❛ ❞❡A ❡stá ❡♠ A✳
▲❡♠❛ ✶✳✶✳✻✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ A ✉♠ ❛t❧❛s C∞ ❞❡ ❞✐♠❡♥sã♦ n ∈ N
♣❛r❛ X✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❛t❧❛s ♠❛①✐♠❛❧ A′ ♣❛r❛
X q✉❡ ❝♦♥té♠ A✳
❉❡♠✳✿ ❇❛st❛ ❝♦♥s✐❞❡r❛rA′
={(U, φ); (U, φ)é ✉♠❛ ❝❛rt❛C∞✲❝♦♠♣❛tí✈❡❧ ❝♦♠ t♦❞❛s
❛s ❝❛rt❛s ❞❡ A }✳ ➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ A′ é ❛t❧❛s ♠❛①✐♠❛❧ ❞❡ X ❡ ❝♦♠té♠A✳
❉❡✜♥✐çã♦ ✶✳✶✳✼✳ ❯♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ s✉❛✈❡ n✲❞✐♠❡♥s✐♦♥❛❧ ♦✉✱ s✐♠✲
♣❧❡s♠❡♥t❡✱ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ n✲❞✐♠❡♥s✐♦♥❛❧ (M,A) é ✉♠ ♣❛r ❢♦r✲
♠❛❞♦ ♣♦r ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❍❛✉s❞♦r✛ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ M ❡ A ✉♠ ❛t❧❛s
♠❛①✐♠❛❧ C∞ ❞❡ ❞✐♠❡♥sã♦
n✳
❙❡✱ ❛❧é♠ ❞✐ss♦✱ M é ❝♦♠♣❛❝t♦ ❡♥tã♦ M é ❝❤❛♠❛❞♦ ❞❡ ✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛✳
❉❡♥♦t❛r❡♠♦s ♣♦r Mn ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ s✉❛✈❡ (M,A) ❞❡ ❞✐♠❡♥sã♦
n✳
❈♦♠♦ t♦❞❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧(M,A)é t❛♠❜é♠ ✈❛r✐❡❞❛❞❡ t♦♣♦❧ó❣✐❝❛✱ s❡✲
❣✉❡✱ ♣❡❧♦ ▲❡♠❛1.1.2✱ q✉❡M é ❧♦❝❛❧♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ❧♦❝❛❧♠❡♥t❡ ❝♦♥❡①♦ ❡ ♠❡tr✐③á✈❡❧
✭❡ ♣♦rt❛♥t♦ ♣❛r❛❝♦♠♣❛❝t♦✮✳
❊①❡♠♣❧♦ ✶✳✶✳✽✳ ❖ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦Rné ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦
n✱ ♣♦✐s é ❝❧❛r❛♠❡♥t❡ ❍❛✉s❞♦r✛✱ ♣♦ss✉✐ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ ❡ ♦ ❛t❧❛s ♠❛①✐♠❛❧ ❝♦♥s✐❞❡✲
r❛❞♦ é ♦ q✉❡ ❝♦♥té♠ ♦ ❛t❧❛s {idRn}✳ ❉❡ ❢♦r♠❛ ❣❡r❛❧✱ q✉❛❧q✉❡r s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ A
❞❡ Rn é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ n✱ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ❛t❧❛s
♠❛①✐♠❛❧ q✉❡ ❝♦♥té♠ {i:A−→Rn}✱ ♦♥❞❡ i é ❛ ❢✉♥çã♦ ✐♥❝❧✉sã♦✳
❊①❡♠♣❧♦ ✶✳✶✳✾✳ ❙❡❥❛♠ ❛s ❢✉♥çõ❡s φ±
i : H
±
i −→ B1(0) ❝♦♠♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✳✹✳
❊♥tã♦{φ±i :H
±
i −→B1(0)} é ✉♠ ❛t❧❛sC∞ ♣❛r❛Sn ❡ ❛ss✐♠ Sn ♣♦ss✉✐ ❡str✉t✉r❛ ❞❡
✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ n✲❞✐♠❡♥s✐♦♥❛❧✳
❊①❡♠♣❧♦ ✶✳✶✳✶✵✳ ❙❡❥❛♠ Mm ❡ Nn ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❱❛♠♦s ❞❡✜♥✐r ✉♠
❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ m + n ♥♦ ❡s♣❛ç♦ ♣r♦❞✉t♦ M ×N✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❝♦❧❡çã♦ ❞❡
t♦❞♦s ♦s ❤♦♠❡♦♠♦r✜s♠♦s ❞❛ ❢♦r♠❛ x×y : U ×V −→ Rn+m✱ ❝♦♠ (x, U) ❡ (y, V)
❝❛rt❛s ❞❡ M ❡ N r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❡✜♥✐❞♦s ♣♦r (x×y)(z, w) = (x(z), y(w))✳ ❈♦♠♦ (x1 ×y1)◦(x×y)−1 = (x1 ◦x−1)×(y1 ◦y−1)✱ s❡❣✉❡ q✉❡ t❛❧ ❝♦❧❡çã♦ é ❛t❧❛s ♣❛r❛
M ×N ❡ ❡st❡ ❡s♣❛ç♦ é ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦ m+n✳
P♦❞❡♠♦s ❣❡♥❡r❛❧✐③❛r t❛❧ r❡s✉❧t❛❞♦ ❝♦♥s✐❞❡r❛♥❞♦r✈❛r✐❡❞❛❞❡sVn1
1 , . . . , Vrnr ❡ ♦❜t❡♠♦s✱
❛♥❛❧♦❣❛♠❡♥t❡✱ ✉♠❛ ❡str✉t✉r❛ ❞❡ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❛r❛ V1 × . . .× Vr✱ ❝♦♠
❞✐♠❡♥sã♦ n1+. . .+nr
✶✳✶✳✶ ❋✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ P❧❛♥♦ ❚❛♥❣❡♥t❡
❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❙❡❥❛♠Mm❡Nn✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡f :M −→N ❢✉♥çã♦
❞❡ M ❡♠ N✳ ❊♥tã♦ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ p0 ∈M✱ s❡ ❡①✐st❡♠ ✉♠❛ ❝❛rt❛ (U, x) ❞❡
M ❡(U′, y) ❝❛rt❛ ❞❡ N t❛✐s q✉❡✿ p
0 ∈U ✱ f(U)⊂U′ ❡ y◦f◦x−1 é ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡
❝❧❛ss❡ C∞ ❡♠ x(p
0)✳
❙❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ M✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ f é ❞✐❢❡✲
r❡♥❝✐á✈❡❧✳
❙❡ ♣❛r❛ ❝❛❞❛ p∈M ❡①✐st❡♠ (U, x) ❡ (U′
, y) ❝❛rt❛s ❞❡ M ❡ N r❡s♣❡❝t✐✈❛♠❡♥t❡✱
t❛✐s q✉❡ y◦f ◦x−1 é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❞❡ ❝❧❛ss❡ Ck,1 ≤ k ≤ ∞✱ ❡♥tã♦ f é ❞✐t❛ s❡r
✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ Ck✳
❆ ❛♣❧✐❝❛çã♦y◦f◦x−1✱ ❞❡✜♥✐❞❛ ❛❝✐♠❛✱ é ❛ ❡①♣r❡ssã♦ ❞❡ f ♥❛s ❝♦♦r❞❡♥❛❞❛s
x ❡ y✱ ❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦rfx,y✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❛♦ ❢❛❧❛r♠♦s ✧❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧✧ ✜❝❛ s✉❜❡♥t❡♥❞✐❞♦ s❡r ❞✐❢❡✲ r❡♥❝✐á✈❡❧ ❞❡ ❝❧❛ss❡ C∞✳
❙❡❥❛♠ X ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦n ❡p∈X✳ ❈♦♥s✐❞❡r❡Cp ={f :J −→X❀J é
✐♥t❡r✈❛❧♦ ❛❜❡rt♦ ❞❡R✱0∈J ❡f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠0❝♦♠f(0) =p}✳ ❈♦♥s✐❞❡r❡ ✉♠❛
❝❛rt❛ (U, x) ❞❡X✱ p ∈U✳ ❊♥tã♦ ♣❛r❛ t♦❞❛ ❝✉r✈❛ f ∈Cp✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦ J′ ⊂J
t❛❧ q✉❡f(J′)⊂U ❡x◦f|
J′ é ✉♠❛ ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠0✳ ❆❧é♠ ❞✐ss♦ (x◦f|J′)′(0)
♥ã♦ ❞❡♣❡♥❞❡ ❞❡J′✱ ❛ss✐♠ ❞❡♥♦t❛r❡♠♦s t❛❧ ❝✉r✈❛✱ s✐♠♣❧❡s♠❡♥t❡ ♣♦r
x◦f✳
❉❡✜♥✐♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♠Cp ❞❛❞❛ ♣♦r✿ f ∼g s❡✱ s♦♠❡♥t❡ s❡✱
(x◦f)′(0) = (x◦g)′(0)✱ ♣❛r❛ ❝❡rt❛ ❝❛rt❛ (U, x) ❞❡ X✳ ➱ ❢❛❝✐❧ ✈❡r q✉❡✱ ✐st♦ ✐♠♣❧✐❝❛
(y◦f)′(0) = (y◦g)′(0)✱ ♣❛r❛ t♦❞❛ ❝❛rt❛(V, y)❞❡X ❡ ♣♦rt❛♥t♦ ∼é ✉♠❛ r❡❧❛çã♦ ❞❡
❡q✉✐✈❛❧ê♥❝✐❛✳ ❖ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡ Cp/∼ é ♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ ❞❡ X ♣❛ss❛♥❞♦ ♣♦r p✱
❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦rTpX✳
P❛r❛ ❝❛❞❛ (U, x) ❝❛rt❛ ❞❡ X✱ p ∈ U✱ ❛ ❢✉♥çã♦ ΨU : TpX −→ Rn ✱ ΨU([f]) =
(x◦f)′(0)❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ é ❜✐❥❡t♦r❛ ❡ ❛ss✐♠T
pX t❡♠ ♥❛t✉r❛❧♠❡♥t❡ ✉♠❛ ❡str✉t✉r❛
❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦n ❡ ΨU é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❆s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡
♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r sã♦ ❞❛❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r✿
• [f] + [g] = Ψ−1
U (ΨU([f]) + ΨU([g])),∀f, g∈Cp❀
• α[f] = Ψ−1
U (αΨU([f])),∀α∈R,∀f ∈Cp✳
❆s ♦♣❡r❛çõ❡s ❞❡✜♥✐❞❛s ❛❝✐♠❛ ✐♥❞❡♣❡♥❞❡♠ ❞❛ ❡s❝♦❧❤❛ ❞❛ ❝❛rt❛ (U, x)✳
❙✉♣♦♥❤❛ f : Mm −→ Nn ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ p∈ M✳ ❙❡❥❛♠ (U, x) ❡ (V, y) ❝❛rt❛s
❞❡Mm ❡Nnr❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s q✉❡f(U)⊂V✳ ❊♥tã♦ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛
TpM
ΨU
f′(p)
/
/Tf(p)N
ΨV
Rm f′
x,y(x(p))
/
/Rn
❉❡ ❢❛t♦✿ ♣❛r❛ t♦❞♦ [λ]∈TpM✱ t❡♠♦s q✉❡✿
ΨV ◦f′(p)([λ]) = ΨV([f ◦λ]) = (y◦f ◦λ)′(0) =
= (fx,y◦(x◦λ))′(0) =fx,y′ (x(p))◦ΨU([λ]).
❉❡✜♥✐çã♦ ✶✳✶✳✶✷✳ ❙❡ f : M −→ N é ✉♠❛ ❢✉♥çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ p ∈ M✱ f′(p) : T
pM −→ Tf(p)N✱ ❞❛❞❛ ♣♦r f′(p) [g] = [f ◦g]✱ é ❛ tr❛♥s❢♦r♠❛çã♦
❞❡r✐✈❛❞❛ ❞❡ f ♥♦ ♣♦♥t♦ p✳ ❚❛❧ ❛♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ é ❧✐♥❡❛r✳
❙❡ f′
(p) é ✐♥❥❡t♦r❛ ♣❛r❛ t♦❞♦ p ❡♠ M✱ ❡♥tã♦ f é ❞✐t❛ s❡r ✉♠❛ ✐♠❡rsã♦✱ ❡
s❡✱ ❛❧é♠ ❞✐ss♦✱ f é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡ ❛ ✐♠❛❣❡♠✱ ❡♥tã♦ f é ❝❤❛♠❛❞❛ ❞❡
♠❡r❣✉❧❤♦✳
❈♦♠♦ ♦ ❞✐❛❣r❛♠❛ ❛♥t❡r✐♦r ❝♦♠✉t❛✱ t❡♠♦s q✉❡ s❡ f é ✉♠❛ ✐♠❡rsã♦ ❡♥tã♦ f′
x,y(x(p)) é ✐♥❥❡t♦r❛✱ ♣❛r❛ t♦❞♦ p ∈ U ⊂ M✱ (U, x) ❝❛rt❛ ❞❡ M✱ (V, y) ❝❛rt❛ ❞❡
N✱ t❛✐s q✉❡f(U)⊂V✳
❖s ❞♦✐s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ ❡♥✉♥❝✐❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ sã♦ ♦s ❢❛♠♦s♦s t❡♦r❡♠❛s ❞❡ ✐♠❡rsã♦ ❡ ♠❡r❣✉❧❤♦ ❞❡ ❲❤✐t♥❡②✿
❚❡♦r❡♠❛ ✶✳✶✳✶✸ ✭❚❡♦r❡♠❛ ❞❡ ✐♠❡rsã♦ ❞❡ ❲❤✐t♥❡②✳✮✳ ❙❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡✲
r❡♥❝✐á✈❡❧ s✉❛✈❡ ❞❡ ❞✐♠❡♥sã♦ n >1✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ ✐♠❡rsã♦ φ:M −→R2n−1✳
❚❡♦r❡♠❛ ✶✳✶✳✶✹ ✭❚❡♦r❡♠❛ ❞❡ ♠❡r❣✉❧❤♦ ❞❡ ❲❤✐t♥❡②✳✮✳ ❙❡ M é ✉♠❛ ✈❛r✐❡❞❛❞❡
❞✐❢❡r❡♥❝✐á✈❡❧ s✉❛✈❡ ❞❡ ❞✐♠❡♥sã♦ n✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ♠❡r❣✉❧❤♦ ψ :M −→R2n✳
P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡s t❡♦r❡♠❛s ✈✐❞❡ [✶✷]✳
❯♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❞✐♠❡♥sã♦né ✉♠ s✉❜❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦S ❞❡Rm(n≤m)
t❛❧ q✉❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ p ∈ S ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛ç❛ V ❞❡ p ❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦
❞✐❢❡r❡♥❝✐á✈❡❧ γ : U −→ V✱ ❞❡ U ❛❜❡rt♦ ❞❡ Rn✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦
❞❡r✐✈❛❞❛ γ′(x) :Rn −→Rm é ✐♥❥❡t♦r❛ ♣❛r❛ t♦❞♦ x∈U✳
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ▼❡r❣✉❧❤♦ ❞❡ ❲❤✐t♥❡②✱ t❡♠♦s q✉❡ t♦❞❛ ✈❛r✐❡❞❛❞❡Xné ♠❡r❣✉✲
❧❤❛❞❛ ❡♠R2n+1✳ ❈♦♥s✐❞❡r❡f ✉♠ t❛❧ ♠❡r❣✉❧❤♦✳ ❊♥tã♦X ❡f(X)sã♦ ❤♦♠❡♦♠♦r❢♦s ❡
f(X)é ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❞✐♠❡♥sã♦n✱ ♣♦✐s ❞❛❞♦p∈f(X)✱ ❡s❝♦❧❤❛(U, φ)❝❛rt❛ ❞❡X
❞❡ t❛❧ ❢♦r♠❛ q✉❡p∈f(U) ✭f(U)é ❛❜❡rt♦ ❡♠ f(X)✱ ♣♦✐s f é ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡
❛ ✐♠❛❣❡♠✮✱ ❡♥tã♦ f ◦φ−1 : φ(U) −→ f(U) é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ (f ◦φ)′
(w) é ✐♥❥❡t♦r❛
♣❛r❛ t♦❞♦w∈φ(U)✱ ♣♦✐sf é ✐♠❡rsã♦✳
P♦rt❛♥t♦✱ t♦❞❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ♠❡s♠❛ ❞✐♠❡♥sã♦✳
◆♦ss♦ ♦❜❥❡t✐✈♦ s❡rá ❡st✉❞❛r ❝❡rt❛s ✈❛r✐❡❞❛❞❡s ✭❛s ●r❛ss♠❛♥♥✐❛♥❛sG1,n❡G2,n✮✱
q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ s✉❜s❡çã♦ s❡❣✉✐♥t❡✱ ❡ ❡♥❝♦♥tr❛r ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ✐♠❡rsã♦ ❞❡st❛s ✈❛r✐❡❞❛❞❡s ❡♠Rm✳
✶✳✶✳✷ ❆s ✈❛r✐❡❞❛❞❡s ●r❛ss♠❛♥♥✐❛♥❛s
G
k,n❙❡❥❛ Gk,n ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ Rk+n ❞❡ ❞✐♠❡♥sã♦k✳
❱❛♠♦s ❞❡✜♥✐r ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ t❛❧ ❢♦r♠❛ q✉❡Gk,n s❡❥❛ ✉♠❛ ✈❛r✐❡❞❛❞❡
❢❡❝❤❛❞❛ ❞❡ ❞✐♠❡♥sã♦kn✱ q✉❡ sã♦ ❝❤❛♠❛❞❛s ❞❡ ✈❛r✐❡❞❛❞❡s ●r❛ss♠❛♥♥✐❛♥❛s Gk,n✳
P❛r❛ ✐st♦✱ ✈❛♠♦s ✉t✐❧✐③❛r ♦
❚❡♦r❡♠❛ ✶✳✶✳✶✺✳ ❈♦♥s✐❞❡r❡X ✉♠ ❝♦♥❥✉♥t♦✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠❛ ❝♦❧❡çã♦ A= {(Ui, φi), i∈J}✱ Ui ⊂X ❡ φi :Ui −→Rn ❢✉♥çõ❡s ✐♥❥❡t♦r❛s✱ t❛✐s q✉❡ ✿
✶✳ φi(Ui) é ❛❜❡rt♦ ❡♠ Rn✱ ♣❛r❛ t♦❞♦ i∈J❀
✷✳ [
i∈J
Ui =X;
✸✳ ❙❡ Ui ∩Uj 6= ∅ ❡♥tã♦ φi(Ui ∩Uj) ❡ φj(Ui ∩Uj) sã♦ ❛❜❡rt♦s ❞❡ Rn✱ t❛✐s q✉❡
φi◦φ−j1 :φj(Ui∩Uj)−→φi(Ui ∩Uj) é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❞❡ ❝❧❛ss❡ C∞.
❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ t♦♣♦❧♦❣✐❛ ♣❛r❛ X t❛❧ q✉❡ A é ✉♠ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦ n✳
❚❛❧ t♦♣♦❧♦❣✐❛ é ❍❛✉s❞♦r✛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞♦ Ui ❡ Uj✱ ♥ã♦ ❞✐s❥✉♥t♦s✱ ♥ã♦
❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s (zn)✱ ❡♠ φi(Ui ∩Uj)✱ ❝♦♠ zn → z ∈ φi(Ui −Uj) ❡
φj ◦φ−i 1(zn)→z′ ∈φj(Uj −Ui)✳
❉❡♠✳✿ ❉❡✜♥❛ τ ={A⊂X;φi(A∩Ui) é ❛❜❡rt♦ ❡♠ Rn,∀i∈J}✳ ➱ ❢❛❝✐❧ ✈❡r q✉❡ τ
é t♦♣♦❧♦❣✐❛ ♣❛r❛ X✳
❙❡❥❛ Ui s✉❜❝♦♥❥✉♥t♦ ❞❡ X✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ j ∈ J✱ t❡♠♦s q✉❡ φj(Uj ∩Ui) é
❛❜❡rt♦ ❞❡Rn✱ ❛ss✐♠ ❝❛❞❛U
i é ✉♠ ❛❜❡rt♦ ❞❡(X, τ)✳ ❆❧é♠ ❞✐ss♦✱ s❡A⊂Ui é ❛❜❡rt♦✱
❡♥tã♦ φi(A) =φi(A∩Ui) é ❛❜❡rt♦ ❞❡ φi(Ui)✱ ❧♦❣♦ φi é ❛♣❧✐❝❛çã♦ ❛❜❡rt❛✳
❙❡ C ⊂ Rn é ❛❜❡rt♦ ❡♥tã♦✱ φj(φ−i 1(C)∩Uj) = (φj ◦φ−i 1(C))∩φj(Uj), ♣❛r❛
t♦❞♦j ∈J✱ ❞❡ss❡ ♠♦❞♦ φi é ❝♦♥tí♥✉❛✳ ▲♦❣♦✱ A é ✉♠ ❛t❧❛s ❞❡ ❞✐♠❡♥sã♦n ♣❛r❛ X✳
❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ♦✉tr❛ t♦♣♦❧♦❣✐❛ ♣❛r❛ X✱ τ′ ❞❡ t❛❧ ♠♦❞♦ q✉❡
A é ✉♠ ❛t❧❛s
♣❛r❛ X✳ ❊♥tã♦ ❝❛❞❛ Ui ♣❡rt❡♥❝❡ ❛ τ′ ❡ s❡ U ∈ τ′ ❡♥tã♦ U ∩Ui ∈ τ′ ❡ φi(U ∩Ui) é
❛❜❡rt♦ ❡♠Rn✱ ❛ss✐♠τ′ ⊂τ✳
❙✉♣♦♥❤❛ q✉❡A⊂X✱φi(A∩Ui)é ❛❜❡rt♦ ❡♠Rn ♣❛r❛ t♦❞♦j ∈J✳ ❊♥tã♦✱ ❝♦♠♦
A=[
j
φ−1(φ(A∩Uj))✱ s❡❣✉❡ q✉❡ τ ⊂τ′✱ ❡ ♣♦rt❛♥t♦ t❛❧ t♦♣♦❧♦❣✐❛ é ú♥✐❝❛✳
P❛r❛ ♦ r❡st❛♥t❡ ❞❛ ❞❡♠♦♥str❛çã♦ ✈✐❞❡[✶, p.115]✳
P❛r❛ ❝❛❞❛ Y ∈Gk,n✱k✲s✉❜❡s♣❛ç♦ ❞❡Rn+k✱ ❛ss♦❝✐❛♠♦s ✉♠❛ ❝❡rt❛ ♠❛tr✐③ M ∈
Mn+k,k(R)✱ t❛❧ q✉❡ ❛s ❝♦❧✉♥❛s ❞❡st❛ ♠❛tr✐③ ❢♦r♠❛♠ ✉♠❛ ❜❛s❡ ♣❛r❛ Y✳ ❊st❛ ♠❛tr✐③
M é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣ê♥❡❛s ❞❡Y✳
❙❛❜❡♠♦s q✉❡ q✉❛❧q✉❡r ♦✉tr❛ ♠❛tr✐③ M′ ∈ M
n+k,k(R) ♥❡st❛s ❝♦♥❞✐çõ❡s é ❞❛
❢♦r♠❛ M′ =M A✱ ❝♦♠ A∈ GL
k(R) ✭❣r✉♣♦ ❞❛s ♠❛tr✐③❡s r❡❛✐s ✐♥✈❡rsí✈❡✐s ❞❡ ♦r❞❡♠
k✮✳
❉❛❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ α ⊂ {1,2, . . . , n+k}, α = {i1 < i2 < . . . < ik}✱ ❞❡♥♦✲
t❛r❡♠♦s ♣♦r α(M) ❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ k ❢♦r♠❛❞❛ ♣❡❧❛s ❧✐♥❤❛s i1, i2, . . . , ik ❞❡ M ❡
♣❡❧❛s ❝♦❧✉♥❛s ❞❡M✳
■♥❞✐❝❛♠♦s ♣♦r α∗ ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡α ❝♦♠ r❡❧❛çã♦ ❛ {1,2, . . . , n+k}✳
❱❡r✐✜❝❛✲s❡ q✉❡ α(Y A) =α(Y)A ❡ α∗(Y A) = α∗(Y)A, ♣❛r❛ t♦❞❛ ♠❛tr✐③ Y ∈
Mn+k,k(R) ❡ t♦❞❛ ♠❛tr✐③ A ✐♥✈❡rsí✈❡❧ ❞❡ ♦r❞❡♠k✳
P❛r❛ ❝❛❞❛ α ⊂ {1,2, . . . , n+k}✱ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ Uα ∈ Gn,k✱ ❢♦r♠❛❞♦
♣❡❧♦s s✉❜❡s♣❛ç♦s H✱ t❛✐s q✉❡ pα : Rn+k −→ [ei1, ei2, . . . , eik]✱ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧
s♦❜r❡ [ei1, ei2, . . . , eik] ❧❡✈❛ H s♦❜r❡ ❛ ✐♠❛❣❡♠✳ ■st♦ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡ α(Y) é
✐♥✈❡rsí✈❡❧✳
❉❡✜♥✐r❡♠♦s ❜✐❥❡çõ❡s xα : Uα −→ Rnk✱ ♣❛r❛ ❝❛❞❛ α = {i1 < i2 < . . . < ik}✳
■❞❡♥t✐✜❝❛♠♦s Rkn ❝♦♠ Mn,k(R)✳ P♦♠♦s xα(H) = α∗(Y α(Y)−1),∀H ∈ U
α ❡ Y
♠❛tr✐③ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❤♦♠♦❣❡♥❡❛s ❞❡ H✳ ❚❛❧ ❢✉♥çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s s❡ Y′ =Y A✱ ❡♥tã♦ α∗(Y Aα(Y A)−1) =α∗(Y)AA−1α(Y)−1 =x
α(H)✳
❚❛♠❜é♠ t❡♠♦s q✉❡ xα é ❜✐❥❡t♦r❛✳ ❉❛❞♦H ∈Mn,k(R)✱ ❞❡✜♥❛ M ∈Mn+k,k(R)✱
t❛❧ q✉❡ α(M) =Idk ❡ α∗(M) =H✱ ❡♥tã♦ Y✱ ❣❡r❛❞♦ ♣❡❧❛s ❝♦❧✉♥❛s ❞❡ M é ❡❧❡♠❡♥t♦
❞❡Uα ❡ xα(Y) = H✱ ❡ ♣♦rt❛♥t♦xα é s♦❜r❡❥❡t♦r❛✳
❙❡ X, Y ∈ Gk,n✱ ❡♥tã♦ s❡♠♣r❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ♠❛tr✐③❡s M ❡ N q✉❡ r❡♣r❡✲
s❡♥t❛♠ Y ❡ X r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s q✉❡ α(M) =α(N) =Idk ❡ xα(Y) =xα(X)⇒
α∗(M) =α∗(N)✳ ▲♦❣♦ M =N ❡ X =Y✳
❙❛❜❡♠♦s t❛♠❜é♠ q✉❡ {Uα}❝♦❜r❡ X✱ ♣♦✐s t♦❞♦ Y ∈Gk,n t❡♠ ♠❛tr✐③ ❞❡ ❝♦♦r✲
❞❡♥❛❞❛s ❤♦♠♦❣❡♥❡❛s H ❝♦♠ ♣♦st♦ r✳ ❆ss✐♠✱ ❡①✐st❡ α ⊂ {1,2, . . . , n+k}✱ α(H) é
✐♥✈❡rtí✈❡❧✳
❆❣♦r❛ s✉♣♦♥❤❛ α, β s✉❜❝♦♥❥✉♥t♦s ❞❡{1,2, . . . , n+k}✱ Uα ❡ Uβ ♥ã♦ ❞✐s❥✉♥t♦s✳
❆s ❛♣❧✐❝❛çõ❡sα¯ :Mn,k(R)−→Mn+k,k(R) t❛❧ q✉❡ α∗(¯α(W)) = W ❡ α(¯α(W)) = Id✱
❡β :Mn+k,k(R)−→Mk(R), W 7→β(W) sã♦ ❝♦♥tí♥✉❛s✳ ❱❡r✐✜❝❛✲s❡ q✉❡
xα(Uα∩Uβ) = (β◦α¯)−1(GLk)✱ ❧♦❣♦ é ❛❜❡rt♦ ❡♠ Rnk. ❆❧é♠ ❞✐ss♦
xβ ◦x−α1(W) =β
∗
(¯α(W))β(¯α(W))−1,
♦ q✉❡ ♠♦str❛ q✉❡ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❞❡ xβ ◦x−α1 sã♦ ♣♦❧✐♥ô♠✐♦s ❡✱ ♣♦rt❛♥t♦✱
sã♦ ❞❡ ❝❧❛ss❡ C∞✳
❆ss✐♠✱ ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ❡①✐st❡ ✉♠❛ t♦♣♦❧♦❣✐❛ ♣❛r❛Gk,nt❛❧ q✉❡{(Uαxα)}
é ❛t❧❛sC∞✳ ❚❛❧ t♦♣♦❧♦❣✐❛ t❡♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧✱ ♣♦✐s ❝❛❞❛U
α t❡♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧
❡ ❡①✐st❡♠ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡st❡s ❛❜❡rt♦s Uα✳ ❚❛♠❜é♠ t❡♠♦s q✉❡ Gk,n é
❍❛✉s❞♦r✛✱ ♣♦✐s s❡α6=β ❡(Wi)é ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠xα(Uα∩Uβ)✭❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦
s✉❜❝♦♥❥✉♥t♦ ❞❡Mn,k(R)✮✱ ❝♦♠ Wi −→W✱ W ∈xα(Uα−Uβ)✱ ❡♥tã♦ β(¯α(W)) ♥ã♦ é
✐♥✈❡rsí✈❡❧ ❡✱ ❞❡ss❡ ♠♦❞♦✱ [β(¯α(Wi))]−1 ♥ã♦ ♣♦❞❡ ❝♦♥✈❡r❣✐r✱ ❡ xβ ◦x−α1(Wi) ❞✐✈❡r❣❡✳
❆ss✐♠✱ s❡ ❝♦♥s✐❞❡r❛r♠♦s ♦ ❛t❧❛s ♠❛①✐♠❛❧ q✉❡ ❝♦♥té♠ {(Uαxα)}✱ s❡❣✉❡ q✉❡ Gk,n é
✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐♠❡♥sã♦kn✳
P❛r❛ ✈❡r q✉❡ Gk,n é ❝♦♠♣❛❝t♦✱ ❝♦♥s✐❞❡r❡ Vk(Rn+k)✱ ♦ ❝♦❥✉♥t♦ ❞❛s ♠❛tr✐③❡s
(n+k)×k ❝✉❥❛ ❛s ❝♦❧✉♥❛s sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡ψ :Vk(Rn+k)−→Gk,n
t❛❧ q✉❡ψ(H)é ♦ s✉❜❡s♣❛ç♦ ❣❡r❛❞♦ ♣❡❧❛s ❝♦❧✉♥❛s ❞❡ H✳ Pr♦✈❡♠♦s q✉❡ψ é ❝♦♥tí♥✉❛✳
P❛r❛ ❝❛❞❛ α={i1 < i2 < . . . < ik} ⊂ {1, . . . , n+k}✱ ❞❡♥♦t❛♠♦s ♣♦r Vα ♦ ❝♦♥❥✉♥t♦
ψ−1(U
α) ❢♦r♠❛❞♦ ♣♦r t♦❞❛s ❛s ♠❛tr✐③❡s Y ∈ Vk(Rn+k)✱ t❛✐s q✉❡ α(Y) é ✐♥✈❡rtí✈❡❧✳
❈♦♠♦ α : Vk(Rn+k) −→ Mk(R) é ❝♦♥tí♥✉❛ ❡ ♦ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s ✐♥✈❡rsí✈❡✐s é
❛❜❡rt♦✱ s❡❣✉❡ q✉❡Vα é ❛❜❡rt♦✳
P♦rt❛♥t♦✱ ♣❛r❛ ♣r♦✈❛r ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ ψ✱ ❜❛st❛ ♣r♦✈❛r q✉❡ ψ|Vα :
Vα −→ Uα é ❝♦♥tí♥✉❛ ♣❛r❛ t♦❞♦ α = {i1 < i2 < . . . < ik} ⊂ {1, . . . , n+k}✳ ▼❛s
xα ◦ψ|Vα(Y) = α∗α(Y)−1 ♣❛r❛ t♦❞♦ Y ∈ Vα✳ ❆ss✐♠✱ xα◦ψ|Vα é ❝♦♥tí♥✉❛✱ ♦ q✉❡
✐♠♣❧✐❝❛ ♥❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ψ|Vα✱ ♣♦✐sxα é ❤♦♠❡♦♠♦r✜s♠♦✳
❈♦♥s✐❞❡r❡F✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡s ❞❡Vk(Rn+k)❡♠ q✉❡ s✉❛s ❝♦❧✉♥❛s
❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ♦rt♦♥♦r♠❛❧✳ ❊♥tã♦ F é ❝♦♠♣❛❝t♦✱ ♣♦✐s s❡ ❝♦♥s✐❞❡r❛r♠♦s ♦
❤♦♠❡♦♠♦r✜s♠♦✿
Φ :M(k+n),k(R)−→Rk(k+n),
❞❛❞♦ ♣♦r Φ([c1, . . . , ck]) = (c1, . . . , ck)✱ ♦♥❞❡ [c1, . . . , ck] ❞❡♥♦t❛ ❛ ♠❛tr✐③ ❝✉❥❛ ❛s
❝♦❧✉♥❛s sã♦ c1, . . . , ck✳❚❡♠♦s q✉❡ Φ(F) é ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦ ♣♦✐s k(c1, . . . , ck)k ≤k
♣❛r❛ t♦❞❛ ♠❛tr✐③ [c1, . . . , ck] ∈ F✳ P♦rt❛♥t♦ F é ❝♦♠♣❛❝t♦ ❡ ❝♦♠♦ t♦❞♦ s✉❜❡s♣❛ç♦
❡♠ Gk,n ♣♦ss✉✐ ❜❛s❡ ♦rt♦♥♦r♠❛❧✱ s❡❣✉❡ q✉❡Gk,n =ψ(F) ❡ ❛ss✐♠ é ❝♦♠♣❛❝t♦✳
◆♦t❡ q✉❡✱ q✉❛♥❞♦ k = 1✱ G1,n =RPn é ♦ n✲❡s♣❛ç♦ ♣r♦❥❡t✐✈♦ r❡❛❧✳
✶✳✷ ❈♦❤♦♠♦❧♦❣✐❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠
Z
2▲❡♠❜r❛♠♦s q✉❡ ♦ ❛♥❡❧ ❞❡ ❝♦❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦
B✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥♦ ❛♥❡❧ Z2 ❞♦s ✐♥t❡✐r♦s ♠ó❞✉❧♦ ✷✱ é ♦ ❛♥❡❧ ❣r❛❞✉❛❞♦
H∗
(B) =
∞
M
i=0
Hi(B),
❛ s♦♠❛ ❞✐r❡t❛ ❞♦s Z2✲♠ó❞✉❧♦s ❞❡ ❝♦❤♦♠♦❧♦❣✐❛ ✭s✐♥❣✉❧❛r✮ ❞❡ B✱ Hi(B)✱ ♠✉♥✐❞♦ ❞♦
♣r♦❞✉t♦ ❝✉♣ ❡ ❝♦♠ ✉♥✐❞❛❞❡ 1∈H0(B)✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ H∗
(B) sã♦ ❞❛ ❢♦r♠❛
w=w0+w1+. . .+wn,
❝♦♠ wi ∈Hi(B)✳
❙❡ x ∈ Hi(B) ❡ y ∈Hj(B)✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ ❝✉♣ ❞❡ x ❡ y s❡rá ❞❡♥♦t❛❞♦ ♣♦r
xy∈Hi+j(B).
❙❡ f : B −→ B′ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♥tr❡ ❞♦✐s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s B ❡
B′✱ ❡♥tã♦✱ ♣❛r❛ ❝❛❞❛ ✐♥t❡✐r♦ n ≥ 0✱ ♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ❡♠ ❝♦❤♦♠♦✲
❧♦❣✐❛ é ❞❡♥♦t❛❞♦ ♣♦r f∗
n : Hn(B
′) −→ Hn(B)✳ ❚❛✐s ❤♦♠♦♠♦r✜s♠♦s ❞❡✜♥❡♠ ✉♠
❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐sf∗ :H∗(B′)−→H∗(B) ❞❛❞♦ ♣♦r✿
f∗
(w0+w1+. . .+wn) = f0∗(w0) +f1∗(w1) +. . .+fn∗(wn),
❝♦♠ f∗
(1) = 1.
❖❜s❡r✈❡ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦X✱H∗(X)é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✱
♣♦✐s✿
s❡x∈Hi(X,Z
2) ❡y ∈Hj(X,Z2)✱ ❡♥tã♦
xy= (−1)i+jyx=yx.
❆❧é♠ ❞✐ss♦✱ t❡♠♦s ♦ s❡❣✉✐♥t❡
▲❡♠❛ ✶✳✷✳✶✳ ❚♦❞♦ ❡❧❡♠❡♥t♦ w ∈ H∗
(B) ❝♦♠ w0 = 1 t❡♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✳
❉❡♥♦t❛r❡♠♦s t❛❧ ❡❧❡♠❡♥t♦ ♣♦r w✱ ✐st♦ é✿ ww=ww= 1✳
❉❡♠✳✿ ❈♦♥s✐❞❡r❡ w¯0 = 1✱w¯1 =w1 ❡ ❞❡✜♥❛ r❡❝✉rs✐✈❛♠❡♥t❡w¯✱ ♣♦r✿
¯
wn= n−1
X
i=1
(wiw¯n−i) +wn, n >1.
❊♥tã♦w¯∈H∗(B,Z
2), (ww¯)0 = 1 ❡ (ww¯)1 =w1+w1 = 0✳ ❙❡ n >1 ❡♥tã♦
(ww¯)n+1 =
n+1
X
j=1
(wjw¯n+1−j) + ¯wn+1 =
n
X
j=1
(wjw¯n+1−j) +wn+1+ ¯wn+1 =
= ¯wn+1+ ¯wn+1 = 0.
❆ss✐♠ (ww¯)i = 0 ∀i≥1✳ ▲♦❣♦ ww= 1✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ ▲❡♠❛✱ ♦❜t❡♠♦s q✉❡ ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡H∗(B)❢♦r♠❛❞♦
♣♦r ❡❧❡♠❡♥t♦s w ∈ H∗
(B) t❛✐s q✉❡ w0 = 1✱ ♠✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ❝✉♣✱ é ✉♠ ❣r✉♣♦
❛❜❡❧✐❛♥♦✳
P❛r❛ ♦s ♥♦ss♦s ♣r♦♣ós✐t♦s✱ ✈❛❧❡ ❧❡♠❜r❛r ❛ ❡str✉t✉r❛ ❞♦ ❛♥❡❧ ❞❡ ❝♦❤♦♠♦❧♦❣✐❛
H∗(RPn,Z
2) ✭✈✐❞❡ ❬✼❪✮✿
Hi(RPn
,Z2)∼=
Z2, se 0≤i≤n
0, se i > n
❉❡♥♦t❛♥❞♦ ♣♦r a ♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ ❞❡ H1(RPn
,Z2) t❡♠♦s q✉❡ ai = a . . . a
✭♣r♦❞✉t♦ ❝✉♣ ❞❡ i ❢❛t♦r❡s ✐❣✉❛✐s ❛ a✮ é ♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ ❞❡ Hi(RPn
,Z2) ♣❛r❛
t♦❞♦1≤i≤n.
❙❡♥❞♦ Mn ✈❛r✐❡❞❛❞❡ ❝♦♥❡①❛✱ s❛❜❡♠♦s q✉❡✿ Hn(Mn,Z
2)∼=Z2 ❡Hi(Mn,Z2)∼=
0s❡ i > n ✭✈✐❞❡ ❬✹❪✮✳
❆❣♦r❛✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡❝♦r❞❛çã♦ s♦❜r❡ ❍♦♠♦❧♦❣✐❛ ❡ ❈♦❤♦♠♦❧♦❣✐❛ ❝❡❧✉❧❛r ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ Z2✳
❯♠❛ n✲❝é❧✉❧❛ ❛❜❡rt❛ e é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❤♦♠❡♦♠♦r❢♦ ❛ ❜♦❧❛ ❛❜❡rt❛ Bn(0,1) ={x∈Rn,kxk<1}✱ ♣❛r❛ ❝❡rt♦ ✐♥t❡✐r♦n ≥0✳
❯♠ ❈✲❲ ❝♦♠♣❧❡①♦ é ✉♠ ♣❛r (X,{eα})✱ ❢♦r♠❛❞♦ ♣♦r ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦
X ❡ ✉♠❛ ❝♦❧❡çã♦ {eα} ❞❡ ❝é❧✉❧❛s ❛❜❡rt❛s ❝♦♥t✐❞❛s ❡♠ X t❛❧ q✉❡✿
✶✳ S
eα =X;
✷✳ P❛r❛ ❝❛❞❛ α✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ fα : Dn(0,1) −→ X✱ ✭Dn(0,1) é ♦
❢❡❝❤♦ ❞❡Bn(0,1)❡♠ Rn✮✱ t❛❧ q✉❡ ❛ r❡str✐çã♦ f
α|Bn(0,1) é ❤♦♠❡♦♠♦r✜s♠♦ s♦❜r❡
❛ ✐♠❛❣❡♠eα✱n✲❝é❧✉❧❛✱ ❡f(Sn−1)❡stá ❝♦♥t✐❞♦ ❡♠ ✉♠❛ ✉♥✐ã♦ ✜♥✐t❛ ❞❡r✲❝é❧✉❧❛s
❝♦♠ r < n;
✸✳ ❙❡ F ⊂X ❡F ∩eα é ❢❡❝❤❛❞♦ ❡♠eα✱ ♣❛r❛ ❝❛❞❛ eα✱ ❡♥tã♦✱ F é ❢❡❝❤❛❞♦ ❡♠X✳
❉❡♥♦t❛r❡♠♦s ♣♦r Xn ❛ ✉♥✐ã♦ ❞❡ t♦❞❛s ❛s r✲❝é❧✉❧❛s ❞❡ e
α ❝♦♠ r ≤ n❀ Xn é ♦
n✲❡sq✉❡❧❡t♦ ❞♦ ❈✲❲ ❝♦♠♣❧❡①♦ X✳
❉❛❞♦ ❈✲❲ ❝♦♠♣❧❡①♦ X✱ ❛ss♦❝✐❛♠♦s ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s✱ ❞❡♥♦t❛❞♦ ♣♦r C∗(X)✿
. . . //Ci(X) ∂i //Ci−1(X)
∂i−1
/
/Ci−2(X) //. . .
♦♥❞❡ Ci(X) = Hi(Xi, Xi−1) ♣❛r❛ t♦❞♦i≥0 ❡Ci(X) = 0 s❡i <0✳ ❖s ❡❧❡♠❡♥t♦s ❞❡
Ci(X) sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝❛❞❡✐❛s ❡♠ ❳✳
❖ ❤♦♠♦♠♦r✜s♠♦ ∂i é ❞❡✜♥✐❞♦ ♣❡❧❛ ❝♦♠♣♦s✐çã♦ ❞♦s ❤♦♠♦♠♦r✜s♠♦s ❥á ❝♦♥❤❡✲
❝✐❞♦s
Hi(Xi, Xi−1) ∂∗
/
/Hi−1(Xi−1)
π∗
/
/Hi−1(Xi−1, Xi−2)
♦♥❞❡π∗ é ♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ❞❡π:Xi−1 −→(Xi−1, Xi−2)❞❛❞♦ ♣♦rπ(x) = x
❡∂∗ é ♦ ❤♦♠♦♠♦r✜s♠♦ ❝♦♥❡❝t❛♥t❡ ❞❡✜♥✐❞♦ ❞❡ t❛❧ ❢♦r♠❛ q✉❡
. . . //Hi(Xi) π∗
/
/Hi(Xi, Xi−1) ∂∗
/
/Hi−1(Xi−1)
π∗
/
/Hi−1(Xi−1, Xi−2) //. . .
é s❡q✉ê♥❝✐❛ ❡①❛t❛✳
❖ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C∗(X) é ❝❤❛♠❛❞♦ ❞❡ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s ❝❡❧✉❧❛r
❞♦ ❈✲❲ ❝♦♠♣❧❡①♦ X✳
➱ ❝♦♥❤❡❝✐❞♦ q✉❡Cn(X)≈
M
eα∈N
Hi(eα, eα−pα)≈
M
eα∈N
Z2✱ ♦♥❞❡ ❝❛❞❛pαé ♣♦♥t♦
❞❡ ❝❛❞❛ ❝é❧✉❧❛eα ❡N é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛sn✲❝é❧✉❧❛s ❞❡X ✭✈✐❞❡ ❬✷❪ ♣á❣✐♥❛ ✷✻✶✮✳
❆ss✐♠✱ Cn(X) é ✉♠ Z2✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✐❣✉❛❧ à ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡
N ❡ ❝❛❞❛ ❝é❧✉❧❛eα ♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞❛ ❝♦♠♦ ✉♠❛ ❝❛❞❡✐❛ ❞❡X (e′j)j∈N ❝♦♠e′j =eα
s❡j =eα ❡ ej′ = 0 ♣❛r❛ j 6=eα✳
❖ n✲és✐♠♦ Z2✲♠ó❞✉❧♦ ❞❡ ❤♦♠♦❧♦❣✐❛ ❞♦ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C∗(X)é ❞❡✜♥✐❞♦
♣♦r Hc,n(X) = Zn/Bn✱ ♦♥❞❡ Zn =ker(∂n) ✭♦ ♥ú❝❧❡♦ ❞❡ ∂n✮ é ♦ ♠ó❞✉❧♦ ❞♦s ❝✐❝❧♦s
❞❡ X ❡ Bn = Im(∂
n+1) ✭❛ ✐♠❛❣❡♠ ❞❡ ∂n+1✮✳ ❖ ♠ó❞✉❧♦ Hc,n(X) é ❛ n✲és✐♠❛
❤♦♠♦❧♦❣✐❛ ❝❡❧✉❧❛r ❞❡ X
❉❡✜♥✐♠♦sCn(X) =C
n(X)∗✱ ❡s♣❛ç♦ ❞✉❛❧ ❞❡Cn(X) = Hi(Xi, Xi−1)✳ ❖❜t❡♠♦s✱
❡♥tã♦✱ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝♦❝❛❞❡✐❛s✱ ❞❡♥♦t❛❞♦ ♣♦rC∗(X)✿
. . . //Ci(X) δi //Ci+1(X) δi+1
/
/Ci+2(X) //. . .
♦♥❞❡ δi é ♦ ❤♦♠♦♠♦r✜s♠♦ ❞✉❛❧ ❞❡ ∂
i✱ ✐st♦ é✱ δi(f) =f ◦∂i+1✳
❖ ❝♦♠♣❧❡①♦ ❞❡ ❝♦❝❛❞❡✐❛s C∗
(X) é ❝❤❛♠❛❞♦ ❞❡ ❝♦♠♣❧❡①♦ ❞❡ ❝♦❝❛❞❡✐❛s ❝❡✲
❧✉❧❛r ❞❡ X ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ Ci(X) sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦❝❛❞❡✐❛s ❞❡ X✳
❖ n✲és✐♠♦Z2✲♠ó❞✉❧♦ ❞❡ ❝♦❤♦♠♦❧♦❣✐❛ ❞♦ ❝♦♠♣❧❡①♦ ❞❡ ❝♦❝❛❞❡✐❛s C∗(X) é ❞❡✲
✜♥✐❞♦ ♣♦r Hn
c(X) = Zn/Bn✱ ♦♥❞❡ Zn = ker(δn) ✭♦ ♥ú❝❧❡♦ ❞❡ δn✮ é ♦ ♠ó❞✉❧♦ ❞♦s
❝♦❝✐❝❧♦s ❞❡X ❡Bn=Im(δn−1)✭❛ ✐♠❛❣❡♠ ❞❡δn−1✮✳❖ ♠ó❞✉❧♦Hn
c(X)é ❛n✲és✐♠❛
❝♦❤♦♠♦❧♦❣✐❛ ❝❡❧✉❧❛r ❞❡ X✳
P❛r❛ r❡❧❛❝✐♦♥❛r ❤♦♠♦❧♦❣✐❛ ✭r❡s♣✳✱ ❝♦❤♦♠♦❧♦❣✐❛✮ ❝❡❧✉❧❛r ❝♦♠ ❤♦♠♦❧♦❣✐❛ ✭r❡s♣✳✱ ❝♦❤♦♠♦❧♦❣✐❛✮ s✐♥❣✉❧❛r✱ ❢❛③❡♠♦s ✉s♦ ❞♦
❚❡♦r❡♠❛ ✶✳✷✳✷✳ ❙❡❥❛ X ✉♠ ❈✲❲ ❝♦♠♣❧❡①♦✳ ❖ n✲és✐♠♦ Z2✲♠ó❞✉❧♦ ❞❡ ❤♦♠♦❧♦✲
❣✐❛ ❞♦ ❝♦♠♣❧❡①♦ C∗(X) é ✐s♦♠♦r❢♦ à n✲és✐♠❛ ❤♦♠♦❧♦❣✐❛ ✭s✐♥❣✉❧❛r✮ ❞❡ X✱ Hn(X)✳
❆♥❛❧♦❣❛♠❡♥t❡✱ Hn
c(X) é ✐s♦♠♦r❢♦ à n✲és✐♠❛ ❝♦❤♦♠♦❧♦❣✐❛ ✭s✐♥❣✉❧❛r✮ ❞❡ X✱ Hn(X).
❉❡♠✳✿ ❬✷❪ ♣á❣✐♥❛ ✷✻✸✳
❈❛♣ít✉❧♦ ✷
❋✐❜r❛❞♦s
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ✜❜r❛❞♦ ❡ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐✲ ❡❞❛❞❡s✳
❉❡✜♥✐çã♦ ✷✳✵✳✸✳ ❯♠ ✜❜r❛❞♦ η = (E, B, p, F) é ❢♦r♠❛❞♦ ♣♦r✿
✶✳ três ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✿ E ✭❡s♣❛ç♦ t♦t❛❧✮✱ B ✭❡s♣❛ç♦ ❜❛s❡✮ ❡ F ✭❡s♣❛ç♦
✜❜r❛✮❀
✷✳ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡ s♦❜r❡❥❡t✐✈❛p:E −→B❀
t❛✐s q✉❡✿ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❞❡ ❛❜❡rt♦s ❞❡ B✱ {Ui}i∈J✱ ♦♥❞❡ ❝❛❞❛ Ui é ✉♠ ❛❜❡rt♦
❧♦❝❛❧♠❡♥t❡ tr✐✈✐❛❧ ❞♦ ✜❜r❛❞♦η✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ψi :p−1(Ui)−→
Ui×F ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ é ❝♦♠✉t❛t✐✈♦✿
p−1(U
i) p
ψi
/
/Ui×F
π1
y
y
Ui
♦✉ s❡❥❛✿ π1◦ψi(x) = p(x),∀x ∈p−1(Ui)✱ ❝♦♠ π1 : Ui×F −→ Ui s❡♥❞♦ ❛ ♣r♦❥❡çã♦
✉s✉❛❧ ♥❛ ♣r✐♠❡✐r❛ ❝♦♦r❞❡♥❛❞❛✳
❙❡❥❛♠ η = (E, B, p, F) ✉♠ ✜❜r❛❞♦✱ {Uj}j∈J ✉♠❛ ❝♦❜❡rt✉r❛ ❞❡ ❛❜❡rt♦s ❧♦❝❛❧✲
♠❡♥t❡ tr✐✈✐❛✐s ❞❡η ❝♦♠ ❤♦♠❡♦♠♦r✜s♠♦sψi :p−1(Ui)−→Ui×F ❡ψj :p−1(Uj)−→
Uj×F t❛✐s q✉❡ π1◦ψi(x) =p(x)✱ ♣❛r❛ t♦❞♦ x∈p−1(Ui) ❡π1◦ψj(x′) =p(x′)✱ ♣❛r❛
t♦❞♦x′ ∈
p−1(U
j)✱ ❝♦♠♦ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✳
❊♥tã♦✱ ❝♦♠♦ p−1(U
i∩Uj) = p−1(Ui)∩p−1(Uj)❡ ✉t✐❧✐③❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡
ψi ❡ψj s❡❣✉❡ q✉❡ψi(p−1(Ui∩Uj)) = (Ui∩Uj)×F ❡ψj(p−1(Ui∩Uj)) = (Ui∩Uj)×F✳
❆❧é♠ ❞✐ss♦✱ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ ❝♦♠✉t❛✿
(Ui∩Uj)×F ψ−1
i
/
/
π1
(
(
p−1(U
i∩Uj) ψj
/
/
p
(Ui∩Uj)×F
π1
v
v
Ui∩Uj
❉❡ ❢❛t♦✿ π1(ψj(x)) = p(x)✱ ♣❛r❛ t♦❞♦x∈p−1(Ui∩Uj)❡✱ ❞❛❞♦z ∈(Ui∩Uj)×F✱
t❡♠♦sp◦ψ−1
i (z) =π1(ψi(ψi−1(z))) =π1(z)✳ ❆ss✐♠π1(ψj◦ψi−1(z)) = π1◦ψj◦ψ−i 1(z) =
π1(z),∀z ∈ (Ui∩Uj)×F. ❊♥tã♦ ψj ◦ψi−1 : (Ui ∩Uj)×F −→ (Ui∩Uj)×F é ❞❛
❢♦r♠❛ ψj ◦ψi−1(x, y) = (x, gji(x)(y))♣❛r❛ ❝❡rt❛ ❢✉♥çã♦ gij.
P❛r❛ ❝❛❞❛ x∈Ui∩Uj ❞❡✜♥❛px :F −→(Ui∩Uj)×F ❞❛❞❛ ♣♦r px(y) = (x, y)✱
❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❚❡♠♦s ❡♥tã♦✿
gji(x) =π2◦ψj◦ψ−i 1◦px
Pr♦♣♦s✐çã♦ ✷✳✵✳✹✳ ❙❡❥❛♠ η= (E, B, p, F) ✉♠ ✜❜r❛❞♦✱ {Uj}j∈J ✉♠❛ ❝♦❜❡rt✉r❛ ❢♦r✲
♠❛❞❛ ♣♦r ❛❜❡rt♦s ❧♦❝❛❧♠❡♥t❡ tr✐✈✐❛✐s ❞❡ η✳ ❊♥tã♦✿
✶✳ gii(x) = idF,∀x∈Ui ❡ i∈J❀
✷✳ gij(x) =gji(x)−1,∀x∈Ui∩Uj❀
✸✳ gij(x)◦gjk(x) = gik(x),∀x∈Ui ∩Uj ∩Uk✳
❉❡♠✳✿
✶✳ ❚❡♠♦s gii(x)(y) = π2◦ψiψi−1◦px(y) =y,∀y∈F.✶
✶❊♠ ❛❧❣✉♥s tr❡❝❤♦s✱ ♣♦r s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ♥♦t❛çã♦ ❡ s❡♠ r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦✱ ♦♠✐t✐r❡♠♦s ♦ s✐♥❛❧
❞❡ ❝♦♠♣♦s✐çã♦ ✉s✉❛❧ ❞❡ ❢✉♥çõ❡s✳
✷✳ (gij(x)◦gji(x))(y) = π2◦ψiψj−1◦pxπ2◦ψjψi−1◦px(y)
=π2◦ψiψj−1◦pxπ2(ψjψ−i 1(x, y)) = y✳
❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s (gji(x)◦gij(x))(y) = y✳ ▲♦❣♦ gij(x) =gji(x)−1✳
✸✳ ❙❡❥❛♠Ui, Uj, Uk ❡x∈Ui∩Uj∩Uk✳ ❊♥tã♦✿ (gij(x)◦gjk(x))(y) = (π2◦ψiψj−1◦
px) ◦(π2 ◦ψjψ−k1 ◦px)(y) = π2 ◦ψiψj−1 ◦ pxπ2(ψjψ−k1(x, y)) = π2 ◦ ψiψ−j1 ◦
ψjψk−1(x, y) =gik(x)(y).
✷✳✶ ❋✐❜r❛❞♦ ❝♦♦r❞❡♥❛❞❛
▲❡♠❜r❛♠♦s q✉❡ ✉♠ ❣r✉♣♦ t♦♣♦❧ó❣✐❝♦(G, τ)é ✉♠ ♣❛r ❢♦r♠❛❞♦ ♣♦r ✉♠ ❣r✉♣♦
G ❡ ✉♠❛ t♦♣♦❧♦❣✐❛ τ ♣❛r❛ G t❛❧ q✉❡ ❛s ❛♣❧✐❝❛çõ❡s (x, y) 7→ xy ❡ x 7→ x−1 sã♦
❝♦♥tí♥✉❛s✳
❙❡❥❛♠ G ✉♠ ❣r✉♣♦ t♦♣♦❧ó❣✐❝♦ ❡ F ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❯♠❛ ❛çã♦ Φ :
G×F −→F é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ f ∈F✱ ❡ g, h∈Gt❡♠✲s❡✿
Φ(eG, f) = f (eG é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❡♠G);
Φ(g,Φ(h, f)) = Φ(gh, f)).
❙❡ Φ : G−→ H(F)✱ ✭H(F) ❝♦♥❥✉♥t♦ ❞♦s ❤♦♠❡♦♠♦r✜s♠♦s ❞❡ F✮✱ Φ(g) = Φg✱
é ✐♥❥❡t♦r❛ ✭♦♥❞❡ Φg : F −→ F é ❞❡✜♥✐❞❛ ♣♦r Φg(f) = Φ(g, f)), ❞✐③❡♠♦s q✉❡ é ✉♠❛
❛çã♦ ❡❢❡t✐✈❛ ❞❡ G ❡♠ F✳
❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❙❡❥❛♠ E✱ B, F ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ Φ :G×F −→F ✉♠❛ ❛çã♦✳
❯♠ ✜❜r❛❞♦ ❝♦♦r❞❡♥❛❞❛ η= (E, B, p, F,Φ, G,{Ui, φi}i) é ❢♦r♠❛❞♦ ♣♦r✿
✶✳ ❯♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛p:E −→B s♦❜r❡❥❡t♦r❛ ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ♣r♦❥❡çã♦❀
✷✳ ❯♠❛ ❛çã♦ Φ :G×F −→F ❡❢❡t✐✈❛❀
✸✳ ❯♠❛ ❢❛♠í❧✐❛ {Ui, ψi}i✱ ❢♦r♠❛❞❛ ♣♦r ❛❜❡rt♦s Ui ❞❡ B✱ ❝♦♠ {Ui}i ❝♦❜❡rt✉r❛
❛❜❡rt❛ ❞❡ B✱ ❡ ❤♦♠❡♦♠♦r✜s♠♦s ψi : p−1(Ui)−→Ui ×F✱ t❛✐s q✉❡ ♦ ❞✐❛❣r❛♠❛
❛❜❛✐①♦ é ❝♦♠✉t❛t✐✈♦✿
p−1(U
i) p
ψi
/
/Ui×F
π1
y
y
Ui
❆❧é♠ ❞✐ss♦✱ ♣❛r❛ t♦❞♦ ♣❛r (Ui, Uj) ❞❡ ❛❜❡rt♦s✱ t❛✐s q✉❡ Ui∩Vj 6=∅✱ ❡ ♣❛r❛ t♦❞♦
x∈Ui∩Uj✱ t❡♠♦s q✉❡ ♦ ❤♦♠❡♦♠♦r✜s♠♦ π2◦Φx◦Φ−x1◦px ♣❡rt❡♥❝❡ ❛ Φ(G)✳
❈❛❞❛ ♣❛r (Ui, ψi) ❝♦♠♦ ♥♦ ✐t❡♠ ✸ ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞❡ s✐st❡♠❛
❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛❧ ❞♦ ✜❜r❛❞♦ η✳
P❛r❛ ❝❛❞❛ ♣❛r (i, j)❞❡ í♥❞✐❝❡s✱ t❛✐s q✉❡ Ui∩Vj =6 ∅✱ ❛s ❢✉♥çõ❡sgij :Ui∩Uj −→
G ❞❡✜♥✐❞❛s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ gij(x) é ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❡♠ G t❛❧ q✉❡
π2◦ψj ◦ψi−1◦px(y) = Φ(gij(x), y),∀y∈F,
sã♦ ❝♦♥tí♥✉❛s✳ ❚❛✐s ❢✉♥çõ❡s sã♦ ❝❤❛♠❛❞❛s ❞❡ ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦✳
❉❛❞♦ ✉♠ ✜❜r❛❞♦ ξ✱ ❞❡♥♦t❛♠♦s ♣♦r E(ξ) ❡ B(ξ) ♦s ❡s♣❛ç♦s t♦t❛❧ ❡ ❜❛s❡✱ r❡s✲
♣❡❝t✐✈❛♠❡♥t❡✱ ❞♦ ✜❜r❛❞♦ξ ❡pξ s✉❛ ❢✉♥çã♦ ♣r♦❥❡çã♦✳
❉❡✜♥✐çã♦ ✷✳✶✳✷✳ ❙❡❥❛♠ η ❡ ξ ❞♦✐s ✜❜r❛❞♦s ❝♦♠ ♦ ♠❡s♠♦ ❡s♣❛ç♦ ❜❛s❡ B, ♠❡s♠❛
✜❜r❛ F,♠❡s♠♦ ❣r✉♣♦ t♦♣♦❧ó❣✐❝♦G ❡ ♠❡s♠❛ ❛çã♦ ❡❢❡t✐✈❛Φ✳ ❊♥tã♦✱ ❞✐③❡♠♦s q✉❡ t❛✐s
✜❜r❛❞♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st❡ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ f :E(η)−→E(ξ) t❛❧ q✉❡✿
✶✳ pξ◦f =pη;
✷✳ ❉❛❞♦s(Ui, ψi)❡ (Vj, αj)s✐st❡♠❛s ❧♦❝❛✐s ❞♦ ✜❜r❛❞♦sη ❡ξ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s
q✉❡ Ui ∩Vj é ♥ã♦ ✈❛③✐♦✱ t❡♠♦s q✉❡ ♦ ❤♦♠❡♦♠♦r✜s♠♦ π2 ◦Ψi ◦f ◦α−j1 ◦px
♣❡rt❡♥❝❡ à Φ(G).
