❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❯❜❡r❧â♥❞✐❛
❋❛❝✉❧❞❛❞❡ ❞❡ ▼❛t❡♠át✐❝❛
▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛
❍❖▼❖▲❖●■❆ ❙■◆●❯▲❆❘
●❛❜r✐❡❧ ❙❛♥t♦s ❞❛ ❙✐❧✈❛
●❛❜r✐❡❧ ❙❛♥t♦s ❞❛ ❙✐❧✈❛
❍❖▼❖▲❖●■❆ ❙■◆●❯▲❆❘
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ à ❈♦♦r❞❡♥❛çã♦ ❞♦ ❈✉rs♦ ❞❡ ▼❛t❡♠át✐❝❛ ❝♦♠♦ r❡q✉✐✲ s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▲✐❝❡♥❝✐❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛✳
❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ▲✐❣✐❛ ▲❛ís ❋ê♠✐♥❛
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ q✉❡ ♣❡r♠✐t✐✉ q✉❡ t✉❞♦ ✐ss♦ ❛❝♦♥t❡❝❡ss❡ ❛♦ ❧♦♥❣♦ ❞❡ ♠✐♥❤❛ ✈✐❞❛✱ ❡ ♥ã♦ s♦♠❡♥t❡ ♥❡st❡s ❛♥♦s ❝♦♠♦ ✉♥✐✈❡rs✐tár✐♦✱ ♠❛s q✉❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s é ♦ ♠❛✐♦r ♠❡str❡ q✉❡ ❛❧❣✉é♠ ♣♦❞❡ ❝♦♥❤❡❝❡r✳
❆♦s ♠❡✉s ♣❛✐s✱ ▲ú❝✐❛ ❡ ❙í❧✈✐♦✱ q✉❡ s❡♠♣r❡ ❢♦r❛♠ ♠✐♥❤❛ ♠❛✐♦r ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ❡ ❢♦rç❛✳ ❙♦✉ ❣r❛t♦ ❛♦ ♠❡✉ ✐r♠ã♦✱ ♣♦r ❛❝r❡❞✐t❛r ❡ ❛♣♦✐❛r ♠❡✉ s♦♥❤♦ ❡ q✉❡ ♥♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ❛✉sê♥❝✐❛ ❞❡❞✐❝❛❞♦s ❛♦ ❡st✉❞♦ s✉♣❡r✐♦r✱ s❡♠♣r❡ ❢❡③ ❡♥t❡♥❞❡r q✉❡ ♦ ❢✉t✉r♦ é ❢❡✐t♦ ❛ ♣❛rt✐r ❞❛ ❝♦♥st❛♥t❡ ❞❡❞✐❝❛çã♦ ♥♦ ♣r❡s❡♥t❡✳ ❊ ❛♦s ♠❡✉s t✐♦s✱ ♣♦r t❡r❡♠ ♠❡ ❛❝♦❧❤✐❞♦ ❞❡ ❜r❛ç♦s ❛❜❡rt♦s ❡♠ s✉❛ ❝❛s❛✳
❆♦s ♠❡✉s ❣r❛♥❞❡s ❛♠✐❣♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❆♥❞ré✱ ❆♠❛♥❞❛✱ ❇r✉♥♦✱ ❊❧✐s✱ ❋❛❜rí❝✐♦✱ ❚❤❛✐❛♥❡ ❡ ❙❛♠❛♥t❤❛✱ q✉❡ s❡♠♣r❡ ❡st✐✈❡r❛♠ ❛♦ ♠❡✉ ❧❛❞♦✱ ❡ q✉❡ ❤♦❥❡ sã♦ ♠❛✐s q✉❡ ❛♠✐❣♦s✱ sã♦ ♠❡✉s ✐r♠ã♦s✳ ❆❣r❛❞❡ç♦ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞❛ ❋❆▼❆❚ q✉❡ ❛❧é♠ ❞❡ ❡♥s✐♥❛❞♦r❡s✱ ❢♦r❛♠ ❣r❛♥❞❡s ✐♥❝❡♥t✐✲ ✈❛❞♦r❡s ❡ ❛♠✐❣♦s✱ q✉❡ ✜③❡r❛♠ t♦❞❛ ❛ ❞✐❢❡r❡♥ç❛ ♥❡ss❡ ♣❡rí♦❞♦✳
❆♦s ♣r♦❢❡ss♦r❡s✱ ▼❛r✐s❛ ❞❡ ❙♦✉s❛ ❡ ❉❛♥✐❡❧ ❈❛r✐❡❧❧♦ q✉❡ ❢♦r❛♠ ♠❡✉s ♦r✐❡♥t❛❞♦r❡s ❡♠ ✐♥✐❝✐❛çõ❡s ❝✐❡♥tí✜❝❛s ❡ ♥✉♥❝❛ ❞❡✐①❛r❛♠ ❞❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♠✳
❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ▲í❣✐❛✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♥✜❛♥ç❛✱ ❡♥♦r♠❡ ✐♥❝❡♥t✐✈♦✱ ♣❛❝✐ê♥❝✐❛ ♥❡ss❛ ♦r✐❡♥✲ t❛çã♦ ❡ ❛❝✐♠❛ ❞❡ t✉❞♦ ♣❡❧❛ ❛♠✐③❛❞❡✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ❛❧❣✉♠❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❛ ár❡❛ ❞❡ t♦♣♦❧♦❣✐❛ ❛❧❣é❜r✐❝❛✱ ❝♦♠ ♦ ❡♥❢♦q✉❡ ❡♠ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r✳ ❆♣r❡s❡♥t❛r❡♠♦s ✉♠ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ▼❛②❡r✲❱✐❡t♦r✐s✱ ❡ ❝❛❧❝✉❧❛r❡♠♦s ❝♦♠ ❡ss❛ ❢❡rr❛♠❡♥t❛ ♦s ❣r✉♣♦s ❞❡ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ❞❡ ❛❧❣✉♥s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿
á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ✸
✷✳✶ ❙❡q✉ê♥❝✐❛ ❡①❛t❛ ❡ s❡♠✐✲❡①❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷✳✷ ❈♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✷✳✸ ❆♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✷✳✹ ❍♦♠♦♠♦r✜s♠♦ ❈♦♥❡❝t❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✸ ❆ ❞❡✜♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛
❙✐♥❣✉❧❛r ✶✸
✸✳✶ ❙✐♠♣❧❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✸✳✷ ❖♣❡r❛❞♦r ❜♦r❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✸✳✸ ❆ ❤♦♠♦❧♦❣✐❛ ❞♦ ♣♦♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✸✳✹ ❍♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ♥♦ ♥í✈❡❧ ③❡r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✸✳✺ ❍♦♠♦♠♦r✜s♠♦ ■♥❞✉③✐❞♦ ❡♠ ❍♦♠♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✻ ■♥✈❛r✐â♥❝✐❛ t♦♣♦❧ó❣✐❝❛ ❡ ❤♦♠♦tó♣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✹ ❆♣❧✐❝❛çõ❡s ❞❛ s❡q✉ê♥❝✐❛ ❞❡
▼❛②❡r✲❱✐❡t♦r✐s ✷✺
✹✳✶ ❙❡q✉ê♥❝✐❛ ❞❡ ▼❛②❡r✲❱✐❡t♦r✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✹✳✷ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
■♥tr♦❞✉çã♦ ✶
✶✳ ■♥tr♦❞✉çã♦
❆ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ é ✉♠ r❛♠♦ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ ❞❛ ▼❛t❡♠át✐❝❛ q✉❡ ❡stá ♥❛ ✐♥t❡rs❡✲ çã♦ ❞❛ ➪❧❣❡❜r❛ ❡ ❞❛ ●❡♦♠❡tr✐❛ ❡ ♣♦ss✉✐ ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛✱ t❛✐s ❝♦♠♦ ●❡♦♠❡tr✐❛ ❉✐❢❡r❡♥❝✐❛❧✱ ❚❡♦r✐❛ ❞♦s ◆ós ❡ t❛♠❜é♠ ♥❛ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❈♦♠♣✉t❛çã♦ ●rá✜❝❛✳ ❆ss✐♠✱ ♥ã♦ ♦❜st❛♥t❡ ♦ ❢❛t♦ ❞❡ ❛ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ s❡r ✉♠ ❝❛♠♣♦ ❛❜str❛t♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ❡❧❛ t❡♠ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❞❡♥tr♦ ❡ ❢♦r❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ❆ss✐♠✱ ❛♣❧✐❝❛✲s❡ ❛ ✈ár✐❛s ár❡❛s✳
❆ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ ❢♦✐ ❢♦rt❡♠❡♥t❡ ✐♥✢✉❡♥❝✐❛❞❛ ♣❡❧❛ t❡♦r✐❛ ❣❡r❛❧ ❞♦s ❝♦♥❥✉♥t♦s ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ●❡♦r❣ ❈❛♥t♦r✱ ♣♦r ✈♦❧t❛ ❞❡ ✶✽✽✵✱ ❡ ❡s♣❛ç♦s ♠étr✐❝♦s ♣♦r ▼❛✉r✐❝❡ ❋r❡❝❤❡t ❡♠ ✶✾✵✻✳
❊♠❜♦r❛ ❛s ♦r✐❣❡♥s ❤✐stór✐❝❛s ❞❛ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ sã♦ ✉♠ ♣♦✉❝♦ ❞✐❢❡r❡♥t❡s✱ ❛ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ ❡ ❛ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ tê♠ ✉♠ ♦❜❥❡t✐✈♦ ❡♠ ❝♦♠✉♠✿ ❞❡t❡r♠✐♥❛r ❛ ♥❛t✉r❡③❛ ❞♦s ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ♣♦r ♠❡✐♦ ❞❡ ♣r♦♣r✐❡❞❛❞❡s q✉❡ ✈ã♦ ❞❡s❞❡ ✐♥✈❛r✐â♥❝✐❛ ❛té ❤♦♠❡♦♠♦r✜s♠♦s✳
❆ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ ❞❡s❝r❡✈❡ ❛ ❡str✉t✉r❛ ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♣♦r ❛ss♦❝✐❛✲ çã♦ ❝♦♠ ✉♠ s✐st❡♠❛ ❛❧❣é❜r✐❝♦✱ ♥♦r♠❛❧♠❡♥t❡ ✉♠ ❣r✉♣♦ ♦✉ ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ❣r✉♣♦s✳ ❉❡ss❡ ♠♦❞♦✱ ❛ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛ ❝♦♥s✐st❡ ❡♠ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s t♦♣♦❧ó❣✐❝♦s ❛tr❛✈és ❞❡ ♠ét♦❞♦s ♣✉r❛♠❡♥t❡ ❛❧❣é❜r✐❝♦s ❡ ✉♠ ❝♦♥❝❡✐t♦ ✐♠♣♦rt❛♥t❡ é ♦ ❞❡ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r✳ ✭❬✶❪✱ ✶✾✼✽✮✳
❆ t❡♦r✐❛ ❞❡ ❤♦♠♦❧♦❣✐❛ é ✉♠ ❛ss✉♥t♦ ✐♠♣♦rt❛♥t❡ ♥❛ ❚♦♣♦❧♦❣✐❛ ❆❧❣é❜r✐❝❛✱ ♣♦✐s ❢♦r♥❡❝❡ ✉♠ ♠ét♦❞♦ ❞❡ ❛ss♦❝✐❛r ❛ ❝❛❞❛ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ✉♠❛ ❝❛t❡❣♦r✐❛ ❞❡ ❣r✉♣♦s ✭♦✉✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ♠ó❞✉❧♦s✮✱ ❝❤❛♠❛❞♦s ❞❡ ❣r✉♣♦ ❞❡ ❤♦♠♦❧♦❣✐❛ ❞❡ss❡ ❡s♣❛ç♦✱ ❞❡ ♠♦❞♦ q✉❡ ❡s♣❛ç♦s ❤♦♠❡♦♠♦r❢♦s ♣♦ss✉❡♠ ❣r✉♣♦s ❞❡ ❤♦♠♦❧♦❣✐❛ ✐s♦♠♦r❢♦s ✭❬✹❪✱ ✷✵✶✷✮✳
❊st❡ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ t❡♠ ❝♦♠♦ ✜♥❛❧✐❞❛❞❡ ❛❜♦r❞❛r ❝♦♠ ♠❛✐s ♣r♦❢✉♥❞✐❞❛❞❡ ❡ ❞❡t❛❧❤❛♠❡♥t♦ ❛❧❣✉♠❛s ❢❡rr❛♠❡♥t❛s ❞❛ t❡♦r✐❛ ❞❡ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ❡ ❛♣❧✐❝❛çõ❡s✱ ♣r♦♣♦r❝✐♦✲ ♥❛♥❞♦ ✉♠❛ ✈✐sã♦ ❛♠♣❧❛ ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❛ t❡♦r✐❛✳
◆♦ ❝❛♣ít✉❧♦ ✉♠✱ r❡❧❛t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ár❡❛ ❞❡ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ❡ss❡♥❝✐❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❝❛♣ít✉❧♦s ♣♦st❡r✐♦r❡s✱ ❞❡st❛❝❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛✱ ❛♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛ ❡ ❤♦♠♦♠♦r✜s♠♦ ❝♦♥❡❝t❛♥t❡✳ ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ✉t✐❧✐③❛❞❛ é ❬✷❪✳
◆♦ ❝❛♣ít✉❧♦ ❞♦✐s✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r✳ P❛r❛ t❛❧ t❡♦r✐❛ é ♥❡❝❡s✲ sár✐♦ ❛ ✐♥tr♦❞✉çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ s✐♠♣❧❡①♦ ❡ ♦♣❡r❛❞♦r ❜♦r❞♦✳ ◆❡ss❡ ❝❛♣ít✉❧♦ ❡①❡♠♣❧✐✜❝❛♠♦s ❛ ❤♦♠♦❧♦❣✐❛ ❞❡ ✉♠ ♣♦♥t♦✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s q✉❡ ♠♦str❛♠ q✉❡ ❛ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r é ✉♠ ✐♥✈❛r✐❛♥t❡ t♦♣♦❧ó❣✐❝♦ ❡ ❤♦♠♦tó♣✐❝♦✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ❞❡ss❡ ❝❛♣ít✉❧♦ sã♦ ❬✶❪✱ ❬✹❪ ❡ ❬✺❪✳
✷ ■♥tr♦❞✉çã♦
◆♦ ❝❛♣ít✉❧♦ três✱ ❛♣r❡s❡♥t❛♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ▼❛②❡r✲❱✐❡t♦r✐s✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❧❝✉❧❛♠♦s ❛ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ❞❛ S1 ❡ ❞❛ ✜❣✉r❛ ♦✐t♦✳ ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❬✺❪✳
❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ✸
✷✳ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿
á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛
◆❡st❡ ❝❛♣ít✉❧♦✱ r❡❧❛t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ❞❡ ❢✉♥✲ ❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳ ❉❡st❛❝❛♠♦s✱ ❡s♣❡❝✐❛❧✲ ♠❡♥t❡✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s✱ ❛♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛ ❡ ♦ ❤♦♠♦♠♦r✜s♠♦ ❝♦♥❡❝t❛♥t❡✳ ❆ ♣r✐♥❝✐♣❛❧ r❡❢❡rê♥❝✐❛ ✉t✐❧✐③❛❞❛ é ❬✷❪✳
✷✳✶ ❙❡q✉ê♥❝✐❛ ❡①❛t❛ ❡ s❡♠✐✲❡①❛t❛
❊s❝r❡✈❡♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛ ❞❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
. . .−→X −→f Y −→g Z −→. . .
❉❡ss❡ ♠♦❞♦✱ ♣❛r❛ ❝❛❞❛ ♠ó❞✉❧♦ q✉❡ ♥ã♦ ❡st❡❥❛ ♥♦s ❡①tr❡♠♦s ❞❛ s❡q✉ê♥❝✐❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ♦ Y ❞❡st❛ s❡q✉ê♥❝✐❛✱ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦f ❝❤❡❣❛♥❞♦ ❡♠Y ❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦g s❛✐♥❞♦ ❞❡ Y✳ ❉❡✜♥✐♠♦sf ❝♦♠♦ s❡♥❞♦ ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❡♥tr❛❞❛ ❡g❝♦♠♦ s❡♥❞♦ ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ s❛í❞❛ ❞❛ s❡q✉ê♥❝✐❛ ♥♦ ♠ó❞✉❧♦ Y✳
❉❡✜♥✐çã♦ ✶✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❞❡ R✲♠ó❞✉❧♦s é ❝♦♠♦ ❞❡✜♥✐❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ ❞♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❡♥tr❛❞❛ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ Kernel ❞♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ s❛í❞❛ ❡♠ t♦❞♦s ♦s ♠ó❞✉❧♦s✱ ❡①❝❡t♦ ♥♦s ❡①tr❡♠♦s ❞❛ s❡q✉ê♥❝✐❛✳
❉❡✜♥✐çã♦ ✷✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❞❛ ❢♦r♠❛
0−→X −→f Y −→g Z −→0
s❡rá ❝❤❛♠❛❞❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❝✉rt❛✳
❉❡✜♥✐çã♦ ✸✳ ❯♠❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛
. . .−→X −→f Y −→g Z −→. . .
❞❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s✱ é ❞✐t❛ ✉♠❛ s❡q✉ê♥❝✐❛ s❡♠✐✲❡①❛t❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ✐♠❛❣❡♠ ❞♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❡♥tr❛❞❛ ❡stá ❝♦♥t✐❞❛ ♥♦ ❑❡r♥❡❧ ❞♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ s❛í❞❛ ♣❛r❛ t♦❞♦ ♠ó❞✉❧♦✱ ❡①❝❡t♦ ♥♦s ❡①tr❡♠♦s ❞❛ s❡q✉ê♥❝✐❛ ✭Im(f)⊂Ker(g)✮✳
✹ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛
❆ s❡q✉ê♥❝✐❛ é s❡♠✐✲❡①❛t❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❝♦♠♣♦s✐çã♦ g ◦f ❞❡ ❞♦✐s ❤♦♠♦♠♦r✜s♠♦s ❝♦♥s❡❝✉t✐✈♦s f ❡ g ♥❛ s❡q✉ê♥❝✐❛ é ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧✳
❖❜s❡r✈❛çã♦ ✶✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❞❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s é s❡♠✐✲❡①❛t❛✱ ♠❛s ♥❡♠ t♦❞❛ s❡q✉ê♥❝✐❛ s❡♠✐✲❡①❛t❛ é ❡①❛t❛✳
❊①❡♠♣❧♦ ✶✳ ❙❡❥❛♠ A ✉♠ s✉❜♠ó❞✉❧♦ ❞❡ ✉♠ R✲♠ó❞✉❧♦ X✱ ✐st♦ é✱ A ⊂ B ♠❛s A 6= B✱ ❡
i:A−→X ♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❝❧✉sã♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ s❡q✉ê♥❝✐❛
0−→A −→i X −→0
é s❡♠✐✲❡①❛t❛ ♠❛s ♥ã♦ é ❡①❛t❛✳ ❖ ♠ó❞✉❧♦ q✉♦❝✐❡♥t❡Q=X/As❡r✈❡ ❝♦♠♦ ✉♠❛ ♠❡❞✐❞❛ ❞♦ ❞❡s✈✐♦ ❞❛ ❡①❛t✐❞ã♦✳
❉❡✜♥✐çã♦ ✹✳ ❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ s❡♠✐✲❡①❛t❛ ❛r❜✐trár✐❛
C :. . .−→X −→f Y −→g Z −→. . . ❞❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s✱ ♦ ♠ó❞✉❧♦ q✉♦❝✐❡♥t❡
Ker(g)/Im(f)
s❡rá ❝❤❛♠❛❞♦ ♠ó❞✉❧♦ ❞❡r✐✈❛❞♦ ❞❛ s❡q✉ê♥❝✐❛ C ♥♦ ♠ó❞✉❧♦ Y✳
Pr♦♣♦s✐çã♦ ✶✳ ❯♠❛ s❡q✉ê♥❝✐❛ s❡♠✐✲❡①❛t❛ ❞❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ R✲♠ó❞✉❧♦s é ❡①❛t❛ s❡✱ ❡ s♦✲ ♠❡♥t❡ s❡✱ t♦❞♦s ♦s s❡✉s ♠ó❞✉❧♦s ❞❡r✐✈❛❞♦s sã♦ tr✐✈✐❛✐s✳
❉❡♠♦♥str❛çã♦✿ (⇒)❈♦♠♦ ❛ s❡q✉ê♥❝✐❛ é ❡①❛t❛✱ s❡❣✉❡ q✉❡ Ker(g) = Im(f)✳ ❆ss✐♠✱
Ker(g)/Im(f) =Im(f)/Im(f) = 0.
P♦rt❛♥t♦✱ t♦❞♦s ♦s ♠ó❞✉❧♦s ❞❡r✐✈❛❞♦s ❞❛ s❡q✉ê♥❝✐❛ s❡♠✐✲❡①❛t❛ sã♦ tr✐✈✐❛✐s✳
(⇐) P♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡ ♦s ♠ó❞✉❧♦s ❞❡r✐✈❛❞♦s sã♦ tr✐✈✐❛✐s✱ ♦✉ s❡❥❛✱ Ker(g)/Im(f) ={0}✳
❉❡st❛ ❢♦r♠❛✱ Ker(g) = Im(f)✳ P♦rt❛♥t♦✱ ❛ ❡qüê♥❝✐❛ é ❡①❛t❛✳
✷✳✷ ❈♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s
❖s ♠ó❞✉❧♦s ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ s❡♠✐✲❡①❛t❛Csã♦ ✉s✉❛❧♠❡♥t❡ ✐♥❞❡①❛❞♦s ♣♦r ✐♥t❡✐r♦s ❝r❡s❝❡♥t❡s ♦✉ ♣♦r ✐♥t❡✐r♦s ❞❡❝r❡s❝❡♥t❡s✳ ❙❡ ✐♥t❡✐r♦s ❞❡❝r❡s❝❡♥t❡s sã♦ ✉s❛❞♦s ❝♦♠♦ í♥❞✐❝❡s✱ ❛ s❡q✉ê♥❝✐❛ s❡♠✐✲ ❡①❛t❛ C é ❝❤❛♠❛❞❛ ❈♦♠♣❧❡①♦ ❞❡ ❈❛❞❡✐❛s ✭♦✉ s❡q✉ê♥❝✐❛ ❜❛✐①❛✮ ❡ ♦s ❤♦♠♦♠♦r✜s♠♦s ❡♠ C✱ s❡rã♦ ❞❡♥♦t❛❞♦s ♣❡❧♦ sí♠❜♦❧♦ ∂ ❡ ✐♥❞❡①❛❞♦s ❞❛ ♠❡s♠❛ ❢♦r♠❛ ❝♦♠♦ ♥♦s ♠ó❞✉❧♦s✳ ❉❡ss❡ ♠♦❞♦ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C é ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
C:...∂n+2
−→Cn+1
∂n+1
−→Cn ∂n
−→Cn−1
∂n−1
−→...,
❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ✺
❝♦♠ ∂n◦∂n+1 = 0✳
❖s ❡❧❡♠❡♥t♦s ❞❡ Ci sã♦ ❝❤❛♠❛❞♦s ❝❛❞❡✐❛s ✐✲❞✐♠❡♥s✐♦♥❛✐s ❞❡ C ❡ ♦s ❤♦♠♦♠♦r✜s♠♦s ∂i sã♦
❝❤❛♠❛❞♦s ♦♣❡r❛❞♦r❡s ❜♦r❞♦✳
❖ ❑❡r♥❡❧ ❞❡ ∂i é ❞❡♥♦t❛❞♦ ♣♦r Zi(C) ❡ ❝❤❛♠❛❞♦ ♠ó❞✉❧♦ ✐✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❝✐❝❧♦s ❞❡ C✳ ❆
✐♠❛❣❡♠ ❞❡ ∂i+1 ❡♠ Ci é ❞❡♥♦t❛❞❛ ♣♦r Bi(C) ❡ é ❝❤❛♠❛❞❛ ♠ó❞✉❧♦ ✐✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ ❜♦r❞♦s ❞❡
❈✳
❉❡✜♥✐çã♦ ✺✳ ❖ ♠ó❞✉❧♦ q✉♦❝✐❡♥t❡
Hi(C) =Zi(C)/Bi(C) =Ker(∂i)/Im(∂i+1)
✭♠ó❞✉❧♦ ❞❡r✐✈❛❞♦ ❞❡ C ♥♦ ♠ó❞✉❧♦ Ci✮ é ❞❡♥♦♠✐♥❛❞♦ ▼ó❞✉❧♦ ❞❡ ❍♦♠♦❧♦❣✐❛ ✐✲❞✐♠❡♥s✐♦♥❛❧
❞❡ C✳
✷✳✸ ❆♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛s
❉❡✜♥✐çã♦ ✻✳ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s✿C :...∂n+2
−→Cn+1
∂n+1
−→Cn ∂n
−→Cn−1
∂n−1
−→...
D:...∂
′
n+2
−→Dn+1
∂n′+1
−→Dn ∂n′
−→Dn−1
∂n′−1
−→... ❯♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛s f :C−→D✱ é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❤♦♠♦♠♦r✜s♠♦s
f ={fi :Ci −→Di/i∈❩}
❞❡ ♠ó❞✉❧♦s s♦❜r❡ R✱ ✐♥❞❡①❛❞❛ ♣♦r ✐♥t❡✐r♦s n∈❩✱ t❛✐s q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❝♦♠✉t❛t✐✈✐❞❛❞❡
∂n◦fn =fn−1◦∂
′
n
é ✈❡r❞❛❞❡✐r❛ ♥♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦
✲
Cn
Dn
❄ ❄
∂n
∂n′
Cn−1
Dn−1
fn−1
fn
✲
♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n ∈❩.
Pr♦♣♦s✐çã♦ ✷✳ ❙❡❥❛ f : C −→ D ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛s✳ ❊♥tã♦✱ ♦s ❤♦♠♦♠♦r✜s♠♦s fi : Ci −→ Di ❧❡✈❛♠ Zi(C) ❡♠ Zi(D) ❡ Bi(C) ❡♠ Bi(D)✱ ♦✉ ♠❡❧❤♦r✱ ❧❡✈❛♠ Ker(∂i) ♥♦
Ker(∂′
i) ❡ Im(∂i+1) ♥❛ Im(∂
′
i+1).
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ♥♦ q✉❛❧ t♦❞♦s ♦s q✉❛❞r❛❞♦s sã♦ ❝♦♠✉t❛t✐✈♦s
✻ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛
C ✿. . . ∂n+2✲
Cn+1 ✲
∂n+1
Cn ✲
∂n ∂n−1
Cn−1 ✲. . .
D ✿. . . ∂
′
n+2✲
Dn+1 ✲
∂n′+1
Dn ✲
∂n′ ∂
′
n−1
Dn−1 ✲. . .
❄ ❄ ❄
fn+1 fn fn−1
❙❡❥❛ z ∈Ker(∂n)✱ ❡♥tã♦ ∂n(z) = 0✳ ❆ss✐♠✱ t❡♠♦s q✉❡
fn−1(∂n(z)) = 0⇒(fn−1◦∂n)(z) = 0⇒∂
′
n◦fn(z) = 0 ⇒∂
′
(fn(z)) = 0
⇒fn(z)∈Ker(∂
′
n)✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s✉♣♦♥❤❛♠♦s z ∈Im(∂n+1)✱ ❡♥tã♦ ❡①✐st❡ w∈ Cn+1 t❛❧ q✉❡ ∂n+1(w) =z✳ ❉❡st❡
♠♦❞♦✱ t❡♠♦s q✉❡✿
fn(z) =fn(∂n+1(w)) = (fn◦∂n+1)(w) = (∂
′
n+1◦fn+1)(w) =∂
′
n+1(fn+1(w))✳
P♦rt❛♥t♦fn(z)∈Im(∂
′
n+1)✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s ❞❡✜♥✐❞♦ ✉♠ ❤♦♠♦♠♦r✜s♠♦ Hi(f) :Hi(C)−→Hi(D) ❞❛❞♦ ♣♦r Hi(f)(z) =fi(z),
♦♥❞❡ z =z+Zi(C)❡fi(z) =fi(z) +Bi(C)✱ ❞♦s ♠ó❞✉❧♦s ❞❡ ❤♦♠♦❧♦❣✐❛ ✐✲❞✐♠❡♥s✐♦♥❛✐sHi(C)❡
Hi(D)✱ q✉❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ✐✲❞✐♠❡♥s✐♦♥❛❧ ❞❡ f✳
❉❡✜♥✐çã♦ ✼✳ ❖ ❤♦♠♦♠♦r✜s♠♦ ✐❞ê♥t✐❝♦ i:C −→C ❞❡ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C é ❛ ❢❛♠í❧✐❛ i={in:Cn−→Cn/n∈❩}
❞❡ ❤♦♠♦♠♦r✜s♠♦s ✐❞ê♥t✐❝♦s in ❞♦s ♠ó❞✉❧♦s Cn✳
Pr♦♣♦s✐çã♦ ✸✳ ❙❡ i : C −→ C é ♦ ❤♦♠♦♠♦r✜s♠♦ ✐❞ê♥t✐❝♦ ❞❡ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C✱ ❡♥tã♦ Hn(i) é ♦ ❤♦♠♦♠♦r✜s♠♦ ✐❞ê♥t✐❝♦ ❞♦ ♠ó❞✉❧♦ Hn(C), ♣❛r❛ t♦❞♦ n ∈❩.
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ i:C −→C ❡ in:Cn−→Cn✳ ❉❡ss❡ ♠♦❞♦✱
Hn(i) :Hn(C)−→Hn(C)
z 7−→Hn(i)(z) =in(z) = z.
P♦rt❛♥t♦✱ Hn(i) = idHn(C)✳
❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ f : C −→ D ❡ g : D −→ E ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s✳ ❊♥tã♦ ❛ ❢❛♠í❧✐❛
h={gi◦fi :Ci −→Ei/i∈❩}
é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞♦ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛sC ♥♦ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛sE✳ ❊ss❡ ❤♦♠♦♠♦r✜s♠♦ h : C −→ E é ❝❤❛♠❛❞♦ ❝♦♠♣♦s✐çã♦ ❞♦s ❤♦♠♦♠♦r✜s♠♦s f : C −→ D ❡ g : D −→ E✱ ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r
❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ✼
g◦f :C −→E✳
Pr♦♣♦s✐çã♦ ✹✳ ❙❡ f : C −→ D ❡ g : D −→ E sã♦ ❤♦♠♦♠♦r✜s♠♦s ❞❡ ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s✱ ❡♥tã♦
Hn(g◦f) = Hn(g)◦Hn(f)
♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n ∈❩.
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛♠♦s xn ∈Hn(C)✳ ❆ss✐♠✱ xn=xn+Bn(C)✳ ❊♥tã♦
Hn(g◦f)(xn) = (gn◦fn)(xn) +Bn(E) (I)
P♦r ♦✉tr♦ ❧❛❞♦✱
Hn(g)◦Hn(f)(xn) = Hn(g)(Hn(f)(xn)) =Hn(g)(fn(xn) +Bn(D)) =gn(fn(xn) +Bn(D)) =
=gn(fn(xn)) +Bn(E) = (gn◦fn)(xn) +Bn(E)✳ (II)
P♦r (I) ❡(II)✱ t❡♠♦s q✉❡ Hn(g◦f) =Hn(g)◦Hn(f)✳
❉❡✜♥✐çã♦ ✽✳ ❈❤❛♠❛♠♦s ❞❡ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s tr✐✈✐❛❧✱ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s O✱ t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ n∈❩✱ t❡♠✲s❡ q✉❡ On ❂ {0}✳
❉❡✜♥✐çã♦ ✾✳ ❯♠ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧ ❞❡ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C ❡♠ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s D é ♦ ❤♦♠♦♠♦r✜s♠♦ h : C → D t❛❧ q✉❡ hn é ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧ ❞♦ ♠ó❞✉❧♦ Cn
♥♦ ♠ó❞✉❧♦ Dn, ♣❛r❛ t♦❞♦ n∈❩.
Pr♦♣♦s✐çã♦ ✺✳ ❙❡ h : C −→ D é ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧ ❞❡ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C ♥♦ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s D✱ ❡♥tã♦
Hn(h) :Hn(C)−→Hn(D)
é ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧ ♣❛r❛ t♦❞♦ n∈❩✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ z ∈Hn(C)✱ ❡♥tã♦
Hn(h)(z) = hn(z) = 0✳
▲♦❣♦✱ Hn(h) é ♦ ❤♦♠♦♠♦r✜s♠♦ tr✐✈✐❛❧ ♣❛r❛ t♦❞♦ n∈❩✳
❉❡✜♥✐çã♦ ✶✵✳ ❉♦✐s ❤♦♠♦♠♦r✜s♠♦s f, g : C −→ D ❞❡ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s C ❡♠ ✉♠ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s D sã♦ ❞✐t♦s ❤♦♠♦tó♣✐❝♦s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❤♦♠♦✲ ♠♦r✜s♠♦s
h={hn:Cn−→Dn+1/n∈❩}
t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ n∈❩✱
∂n+1◦hn+hn−1◦∂
′
n=fn−gn
✽ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛
❝♦♥❢♦r♠❡ ✐♥❞✐❝❛ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛✿
✲
Cn
Cn−1
❄ ❄
hn−1
hn
fn
gn
Dn+1
Dn
∂n+1
∂n′
✲ ◗
◗ ◗
◗ ◗◗s ◗
◗ ◗
◗ ◗◗s
❆ ❢❛♠í❧✐❛ h ❞❡s❝r✐t❛ ❛❝✐♠❛✱ é ❝❤❛♠❛❞❛ ✉♠❛ ❤♦♠♦t♦♣✐❛ ✭♦✉ ✉♠❛ ❤♦♠♦t♦♣✐❛ ❞❡ ❝❛❞❡✐❛s✮ ❡♥tr❡ ♦s ❤♦♠♦♠♦r✜s♠♦s f ❡ g✳ ❊♠ sí♠❜♦❧♦s
h :f ≃g :C −→D.
Pr♦♣♦s✐çã♦ ✻✳ ❙❡ ❞✉❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❝❛❞❡✐❛s f, g : C −→ D ❞❡ ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s sã♦ ❤♦♠♦tó♣✐❝❛s✱ ❡♥tã♦
Hn(f) =Hn(g) :Hn(C)−→Hn(D)
♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n ∈❩.
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ h : f ≃ g : C −→ D ❡ x ∈ Hn(C)✳ ❙✉♣♦♥❤❛♠♦s z ∈ Zn(C)✱ t❛❧
q✉❡ p(z) =x✱ ♦♥❞❡ p❞❡♥♦t❛ ❛ ♣r♦❥❡çã♦ ♥❛t✉r❛❧ ❞♦ ♠ó❞✉❧♦ Zn(C) s♦❜r❡ s❡✉ ♠ó❞✉❧♦ q✉♦❝✐❡♥t❡
Hn(C)✳ ❊♥tã♦
fn(z)−gn(z) =∂n+1[hn(z)] +hn−1[∂
′
n(z)] =∂n+1[h(z)]
❝♦♠ ∂n′(z) = 0✳ ❈♦♠♦∂[hn(z)]∈Bn(D)✱ t❡♠♦s
[Hn(f)](x) = [Hn(g)](x)
P♦rt❛♥t♦✱ ❝♦♠♦ x é ✉♠ ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦ ❞❡ Hn(C)✱ t❡♠♦s q✉❡ Hn(f) =Hn(g)✳
▲❡♠❛ ✶✳ ❙❡❥❛ 0 −→ C −→f D −→g E −→ 0 ✉♠❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❝✉rt❛ ❞❡ ✉♠ ❝♦♠♣❧❡①♦ ❞❡
❝❛❞❡✐❛s✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ n ∈❩✱ ❛ s❡q✉ê♥❝✐❛
Hn(C) Hn(f)
−→ Hn(D) Hn(g)
−→ Hn(E)
❞❡ R✲♠ó❞✉❧♦s ❞❡ ❤♦♠♦❧♦❣✐❛ é ❡①❛t❛✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ g◦f = 0✱ s❡❣✉❡ q✉❡
Hn(g)◦Hn(f) =Hn(g◦f) = 0✳
❆ss✐♠✱
Im[Hn(f)]⊂Ker[Hn(g)], (I)
❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ✾
♣❛r❛ t♦❞♦ n ∈ ❩✳ ❆❣♦r❛✱ ♣❛r❛ ♣r♦✈❛r♠♦s ❛ ♦✉tr❛ ✐♥❝❧✉sã♦✱ ❝♦♥s✐❞❡r❡♠♦s α ∈ Ker[Hn(g)],
α=z =z+Bn(D);z ∈Zn(D)✱ s❡♥❞♦Zn(D)⊂Dn✳ ❉❡ss❡ ♠♦❞♦✱
[Hn(g)](α) = 0 ⇒[Hn(g)](z) =gn(z) = 0✱
t❡♠♦s q✉❡ gn(z)∈Bn(E) =Im(∂n+1) ❡ ∂nE+1 :En+1 −→En✳
❈♦♥s✐❞❡r❡ ♦ ❞✐❛❣r❛♠❛✿
0 ✲ C
n+1 ✲
fn+1
Dn+1 ✲
gn+1
En+1 ✲ 0
0 ✲ C
n ✲
fn
Dn ✲
gn
En ✲. . .
❄ ❄ ❄
∂C
n+1 ∂Dn+1 ∂nE+1
❈♦♠♦ ❛s ❧✐♥❤❛s sã♦ s❡q✉ê♥❝✐❛s ❡①❛t❛s✱ s❛❜❡♠♦s q✉❡gn+1 é ❡♣✐♠♦r✜s♠♦✳ ❊♥tã♦✱ ❡①✐st❡x∈Dn+1✱
t❛❧ q✉❡ gn+1(x) =y✳ ▲♦❣♦✱
gn(z−∂nD+1(x)) =gn(z)−gn(∂nD+1(x)) =gn(z)−∂nE+1(gn+1(x)) = gn(z)−∂nE+1(y) =
=∂E
n+1(y)−∂nE+1(y) = 0
P♦rt❛♥t♦✱ z−∂D
n+1 ∈ Ker(gn) = Im(fn)✳ ❆ss✐♠✱ ❡①✐st❡ w ∈Cn ❝♦♠ fn(w) = z−∂nD+1(x)✳
❊♥tã♦
fn−1[∂nC(w)] = ∂nD(fn(w)) =∂nD(z−∂nD+1(x)) =∂nD(z) = 0✳
❈♦♠♦ fn−1 é ♠♦♥♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ ∂(w) = 0 ❡ ❛ss✐♠ w∈Ker(∂nC) = Zn(C)✳
❆❣♦r❛✱ s❡❥❛ p❛ ♣r♦❥❡çã♦ ♥❛t✉r❛❧ ❞♦ ♠ó❞✉❧♦ Zn(C)♥♦ s❡✉ ♠ó❞✉❧♦ q✉♦❝✐❡♥t❡ Hn(C)✳
p:Zn(C)−→Zn(C)/Bn(C) =Hn(C)
w7−→p(w) = w∈Hn(C)
❚❡♠♦s q✉❡[Hn(g)](α) = 0✱ ❡♥tã♦α∈Im(Hn(f))✳ ❈♦♠♦α= [Hn(f)](w)❡z =fn(w)✱ ♣♦❞❡♠♦s
❝♦♥❝❧✉✐r q✉❡fn(w)−z ∈Bn(D)✳ ▲♦❣♦✱z =α❡∂nD+1 ∈Bn(D) = Im(∂nD+1)✳ ❉❡ss❡ ♠♦❞♦✱ t❡♠♦s
fn(w) = z−∂nD+1(x)∈α =z =z+Im(∂nD+1)
P♦rt❛♥t♦✱ fn(w) = z ❡Hn(f)(w) = α✳ ❆ss✐♠✱
Ker[Hn(g)]⊂Im[Hn(f)] (II)
P♦r (I) ❡(II)✱ t❡♠♦s q✉❡ Im[Hn(f)] = Ker[Hn(g)]✳
✶✵ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛
✷✳✹ ❍♦♠♦♠♦r❢✐s♠♦ ❈♦♥❡❝t❛♥t❡
❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❞❡ ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s✿
E ✿ 0 ✲ C
n ✲
fn
Dn ✲
gn
En ✲ 0
C ✿ 0 ✲ C
n−1 ✲
fn−1
Dn−1 ✲
gn−1
En−1 ✲0
❄ ❄ ❄
∂n ∂
′ n ∂ ′′ n ❄ ❄ ❄ ❄ ❄ ❄
♦♥❞❡ ❝❛❞❛ ❧✐♥❤❛ é ❡①❛t❛✳ ❙✉♣♦♥❤❛♠♦s z ∈Ker(∂′′
n) ⊂ En✳ ❈♦♠♦ gn é s♦❜r❡❥❡t♦r❛✱ ❡①✐st❡ y ∈ Dn t❛❧ q✉❡ gn(y) = z✳
❚❡♠♦s ❛ss✐♠ q✉❡ ∂′
n(y) ∈ Dn−1✳ ▼♦str❛✲s❡ q✉❡ ∂
′
n(y) ∈ Ker(gn−1) = Im(fn−1)✳ ❆ss✐♠ ∃x ∈
Cn−1 t❛❧ q✉❡ fn−1(x) = ∂
′
n(y)✳ ▼♦str❛✲s❡ q✉❡ x∈Ker(∂n−1)✳
❊♥tã♦✱ ❞❡✜♥✐♠♦s✿
△:Hn(E)−→Hn−1(C)
z =z+Im(∂′′
n+1)7−→ △(z) = x=x+Im(∂n)✳
△ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡ é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳✭❱❡r ▲❡♠❛s ✻✳✻✱ ✻✳✼ ❡ ✻✳✽ ❞❛s ♣á❣✐♥❛s ✹✽ ❡ ✹✾ ❞❛
r❡❢❡rê♥❝✐❛ ❬✷❪✳✮
❊st❡ ❤♦♠♦♠♦r✜s♠♦△:Hn(E)−→Hn−1(C)✱ ❝♦♥str✉í❞♦ ♣❛r❛ ❝❛❞❛ ✐♥t❡✐r♦n✱ s❡rá ❝❤❛♠❛❞♦
❞❡ ❤♦♠♦♠♦r✜s♠♦ ❝♦♥❡❝t❛♥t❡✳
❚❡♦r❡♠❛ ✶✳ ❈♦♥s✐❞❡r❡ ✉♠ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❝✉rt❛
0−→C −→f D−→g E −→0
❞❡ ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s✳ ❊♥tã♦✱ ❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❧♦♥❣❛ ❞❡ ❤♦♠♦❧♦❣✐❛ ...−→Hn(C)
Hn(f)
−→ Hn(D) Hn(g)
−→ Hn(E)
△
−→Hn−1(C)
Hn−1(f)
−→ Hn−1(D)−→...
é ❡①❛t❛✳
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡ Im(Hn(f)) = Ker(Hn(g))✱ ♣❛r❛ t♦❞♦ n ∈ ❩✳ ❘❡st❛ ♠♦str❛r♠♦s
q✉❡
(i)Im(Hn(g)) = Ker(△) ❡ (ii)Im(△) = Ker(Hn−1(f))
(i) ❈♦♥s✐❞❡r❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ❤♦♠♦❧♦❣✐❛
❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛ ✶✶
Hn(D) Hn(g)
−→ Hn(E)
△
−→Hn−1(C)
❙❡❥❛ α∈Im(Hn(g))⊂Hn(E)✳ ❊♥tã♦ α=Hn(g)(β) t❛❧ q✉❡ β ∈Hn(D)✳
❆ss✐♠✱ β = z +Bn(D), z ∈ Zn(D)✳ ▲♦❣♦✱ ∂
′
n(z) = 0✳ ▲♦❣♦✱ ❡①✐st❡ v = 0 ∈ Cn−1 t❛❧ q✉❡
fn−1(v) = 0 = ∂
′
n(z)✳ ❯s❛♥❞♦ ❡st❡ ❢❛t♦ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ △ ♠♦str❛✲s❡ q✉❡ △(α) = 0 ❡ ❛ss✐♠
Im(Hn(g))⊂Ker(△)✳
P❛r❛ ♠♦str❛r♠♦s ❛ ♦✉tr❛ ✐♥❝❧✉sã♦✱ s❡❥❛ α ∈ Ker(△) ⊂ Hn(E)✳ ❆ss✐♠✱ α = z +Bn(E)
t❛❧ q✉❡ z ∈ Zn(E)✳ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ △(α) ❡①✐st❡♠ u ∈ Dn, v ∈ Cn−1 ❝♦♠ gn(u) = z ❡
fn−1(v) = ∂
′
n(u) t❛❧ q✉❡ △(α) = v = v +Bn−1(C) = 0. ❉❡ss❡ ♠♦❞♦✱ v ∈ Bn−1(C) ❡ ❡①✐st❡
w ∈ Cn ❝♦♠ ∂n−1(w) = 0✳ ❙❡❥❛ y = u− fn(w) ∈ Dn ❡ β = y+Bn+1(D)✳ ▼♦str❛✲s❡ q✉❡
Hn(g)(β) = α✳ ❆ss✐♠✱ α∈Im(Hn(g))✳
❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ♣♦❞❡♠♦s ❞❡♠♦♥str❛r (II).
✶✷ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✿ á❧❣❡❜r❛ ❤♦♠♦❧ó❣✐❝❛
❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r ✶✸
✸✳ ❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛
❙✐♥❣✉❧❛r
◆❡st❡ ❝❛♣ít✉❧♦✱ r❡❧❛t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ❣r✉♣♦s ❞❡ ❤♦♠♦❧♦❣✐❛✳ ❉❡st❛❝❛♠♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ ❛ ❞❡✜♥✐çã♦ ❞❡ s✐♠✲ ♣❧❡①♦✱ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❤♦♠♦t♦♣✐❛✳ P❛r❛ ♦ ❧❡✐t♦r ♥ã♦ ❤❛❜✐t✉❛❞♦ ❛ t❡♦r✐❛ ❞❡ ♠ó❞✉❧♦s✱ ♣ré✲r❡q✉❡s✐t♦ ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❝❛♣ít✉❧♦✱ s✉❣❡r✐♠♦s ❬✷❪✳ ❆s ♣r✐♥❝✐♣❛✐s r❡❢❡rê♥❝✐❛s ❞❡ss❡ ❝❛♣ít✉❧♦ sã♦ ❬✶❪✱ ❬✹❪ ❡ ❬✺❪✳
✸✳✶ ❙✐♠♣❧❡①♦
❉❡✜♥✐çã♦ ✶✶✳ ❙❡❥❛♠ a, b∈ ❘n✱ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❝♦♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ a ❡ ♣♦♥t♦ ✜♥❛❧ b é ♦
s✉❜❝♦♥❥✉♥t♦ [a, b] ={(1−t)a+bt,0≤t ≤1} ⊂❘n✳
❉❡✜♥✐çã♦ ✶✷✳ ❖ s✉❜❝♦♥❥✉♥t♦X ⊂❘n é ❝♦♥✈❡①♦ s❡ ♣❛r❛ q✉❛✐sq✉❡ra, b∈X✱ t❡♠♦s[a, b]⊂X✳
Convexo Não convexo
❉❡✜♥✐çã♦ ✶✸✳ ❙❡❥❛ X ⊂ ❘n✱ ❝❤❛♠❛♠♦s ❞❡ ❢❡❝❤♦ ❝♦♥✈❡①♦ ❞❡ X ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s
s✉❜❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦s ❞❡ ❘n q✉❡ ❝♦♥té♠ X✳
❉❡✜♥✐çã♦ ✶✹✳ ❯♠ ♣✲s✐♠♣❧❡①♦ s ❞♦ ❘n é ♦ ❢❡❝❤♦ ❝♦♥✈❡①♦ ❞❡ ✉♠❛ ❝♦❧❡çã♦ ❞❡ (p+ 1) ♣♦♥✲
t♦s {x0, x1, . . . , xp} ♥♦ ❘n✱ ♦♥❞❡ ❝❛❞❛ x1 −x0, . . . , xp −x0 ❢♦r♠❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡❛r♠❡♥t❡
✐♥❞❡♣❡♥❞❡♥t❡✳ ❖s ❡❧❡♠❡♥t♦s x0, x1, . . . , xp sã♦ ❝❤❛♠❛❞♦s ✈ért✐❝❡s ❞❡ s✳
❊①❡♠♣❧♦ ✷✳ ❖s 0✲s✐♠♣❧❡①♦s sã♦ ♦s ♣♦♥t♦s ❞♦ Rn✱ ♦s 1✲s✐♠♣❧❡①♦s sã♦ s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❞♦
Rn✱ ♦s 2✲s✐♠♣❧❡①♦s sã♦ tr✐â♥❣✉❧♦s ❞♦ Rn ❝♦♠ s✉❛ ❢r♦♥t❡✐r❛ ❡ s❡✉ ✐♥t❡r✐♦r ❡ ♦s 3✲s✐♠♣❧❡①♦s sã♦
tr❡tr❛❡❞r♦s ❞♦ Rn ❝♦♠ s✉❛ ❢r♦♥t❡✐r❛ ❡ s❡✉ ✐♥t❡r✐♦r✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈✐s✉❛❧✐③❛r ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳
✶✹ ❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r
0-simplexo 1-simplexo 2-simplexo 3-simplexo
Ix1 x0
Ix2
Ix3
Ix
1
x0
x0 x
0 Ix1
Ix2
Pr♦♣♦s✐çã♦ ✼✳ ❙❡❥❛ x0, x1,· · · , xp ⊂❘n✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿
✐✮ ❖ ❝♦♥❥✉♥t♦ {x1−x0,· · · , xp−x0} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡❀
✐✐✮ ❙❡
p
X
i=0
sixi = p
X
i=0
tixi ❡ p
X
i=0
si = p
X
i=0
ti,
❡♥tã♦ si =ti✱ ♣❛r❛ i= 0,· · · , p.
❉❡♠♦♥str❛çã♦✿
(i)⇒(ii) ❙❡
p
X
i=0
sixi = p
X
i=0
tixi ❡ p
X
i=0
si = p
X
i=0
ti✱ ❡♥tã♦
0 =
p
X
i=0
(si−ti)xi = p
X
i=0
(si−ti)xi−
" p X
i=0
(si−ti)
# x0 = p X i=1
(si−ti)(xi−x0).
P♦r ✐✮ t❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ {x1 −x0, . . . , xp −x0} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❧♦❣♦ si = ti
♣❛r❛ i= 1, . . . , p. ❋✐♥❛❧♠❡♥t❡✱ ✐ss♦ ✐♠♣❧✐❝❛ s0 =t0✱ ❞❡s❞❡ q✉❡
p
X
i=0
si = p
X
i=0
ti.
(ii)⇒(i) ❙❡
p
X
i=1
(ti)(xi−x0) = 0✱ ❡♥tã♦
p
X
i=1
tixi = p
X
i=1
ti
!
x0✱ ♦✉ s❡❥❛✱
t0x0+t1x1+· · ·+tpxp = (t0 +t1+· · ·+tp)x0+ 0x1+· · ·+ 0xp,
s❡♥❞♦t0+t1+· · ·+tp = (t0+t1+· · ·+tp) + 0 +· · ·+ 0✳ ❆ss✐♠ ♣♦r(ii)♦s ❝♦❡✜❝✐❡♥t❡st1, . . . , tn
t♦❞♦s ❞❡✈❡♠ s❡r ③❡r♦✳ ■ss♦ ♣r♦✈❛ ❛ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ❞♦ ❝♦♥❥✉♥t♦ {x1−x0,· · · , xp−x0}✳
❈♦r♦❧ár✐♦ ✶✳ ❙❡ ♦ p✲s✐♠♣❧❡①♦ s é ♦ ❢❡❝❤♦ ❝♦♥✈❡①♦ ❞❡ {x0,· · · , xp} ❡♥tã♦ t♦❞♦ ♣♦♥t♦ ❞❡ s t❡♠
✉♠❛ ú♥✐❝❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ❢♦r♠❛
p
X
i=0
tixi✱ ♦♥❞❡ ti ≥0✱ ♣❛r❛ t♦❞♦ i ❡ Pti = 1✳
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡
s=
( p
X
i=0
tixi ∈❘n
p X i=0
ti = 1 ❡ti ≥0
)
❖ ❝♦♥❥✉♥t♦ s é ❝♦♥✈❡①♦✳
❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r ✶✺
❉❡ ❢❛t♦✱ s❡❥❛♠ p, q ∈ s✱ ❡♥tã♦ p =
p
X
i=0
tixi, q = p
X
i=0
sixi, t❛❧ q✉❡ p
X
i=0
ti = p
X
i=0
si = 1✱
si ≥0, ti ≥0✳
❈♦♥s✐❞❡r❡ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛✱ tp+ (1−t)q,0≤t≤1✱ ❡♥tã♦
tp+ (1−t)q =t
p
X
i=0
tixi
!
+ (1−t)
p
X
i=0
sixi
!
=
p
X
i=0
(tti+ (1−t)si)xi ∈S,
♣♦✐s
p
X
i=0
(tti+ (1−t)si) = t p
X
i=0
ti + (1−t) p
X
i=0
si =t.1 + (1−t).1 = 1 ❡ tti+ (1−t)si ≥0.
◆♦t❡ q✉❡s é ♦ ❢❡❝❤♦ ❝♦♥✈❡①♦ ❞♦ ❝♦♥❥✉♥t♦{x0, x1,· · · , xp}✳ P♦rt❛♥t♦✱ s❡ ♦ p✲s✐♠♣❧❡①♦ ❞❡s é
♦ ❢ê❝❤♦ ❝♦♥✈❡①♦ ❞❡ {x0, x1,· · · , xp} ❡♥tã♦✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ t♦❞♦ ♣♦♥t♦ ❞❡ s t❡♠ ✉♠❛
ú♥✐❝❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ❢♦r♠❛
p
X
i=0
tixi✱ ♦♥❞❡ti ≥0✱ ♣❛r❛ t♦❞♦ i❡ p
X
i=0
ti = 1✳ ✳
❉❡✜♥✐çã♦ ✶✺✳ ❖ p✲s✐♠♣❧❡①♦ σp ❞♦ ❘p+1 ❝♦♠ ✈ért✐❝❡s x0 = (1,0,· · · ,0),
x1 = (0,1,0,· · ·,0),· · · , xp = (0,· · · ,0,1)✱ é ❞❡♥♦♠✐♥❛❞♦ ♦ p✲s✐♠♣❧❡①♦ ♣❛❞rã♦✳
❖❜s❡r✈❛çã♦ ✷✳ ❖ p✲s✐♠♣❧❡①♦ ♣❛❞rã♦σp é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s(t0, t1,· · · , tp)∈❘p+1✱
❝♦♠
p
X
i=0
ti = 1 ❡ ti ≥0✱ ♣❛r❛ ❝❛❞❛ i✳
❊①❡♠♣❧♦ ✸✳ ❖ 0✲s✐♠♣❧❡①♦ ♣❛❞rã♦ σ0 é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦ ❡❧❡♠❡♥t♦ 1∈❘✱ ♦ 1✲s✐♠♣❧❡①♦
♣❛❞rã♦ σ1 é ♦ ❝♦♥❥✉♥t♦
(t0, t1)∈❘2
P
ti = 1, ti ≥0 ❡ ♦ 2✲s✐♠♣❧❡①♦ ♣❛❞rã♦ σ2 é ♦ ❝♦♥❥✉♥t♦
(t0, t1, t2)∈❘3
P
ti = 1, ti ≥0 ✱ ❝♦♠♦ ✈✐s✉❛❧✐③❛❞♦s ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳
0-simplexo padrão
1-simplexo padrão
2-simplexo padrão
ix =10 x0
ix1
x0 Ix2
ix1
❖❜s❡r✈❛çã♦ ✸✳ ◆♦t❡ q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s p✲s✐♠♣❧❡①♦s ❡♠ ❘n✱ ♠❛s p✲s✐♠♣❧❡①♦ ♣❛❞rã♦ ❡①✐st❡
❛♣❡♥❛s ✉♠ ❡ t❡♠ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❘p+1 ❡ ❞✐♠❡♥sã♦ p✳
❉❡✜♥✐çã♦ ✶✻✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ✉♠ p−s✐♠♣❧❡①♦ s✐♥❣✉❧❛r ❡♠ X é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ φ : σp → X✳ ❖s p−s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s sã♦ t❛♠❜é♠ ❝❤❛♠❛❞♦s ❞❡ s✐♠♣❧❡①♦s ❞❡
❞✐♠❡♥sã♦ p✳
❖❜s❡r✈❛çã♦ ✹✳ ❉❡♥♦t❡♠♦s Cn(X) ♦ ❝♦♥❥✉♥t♦ ❞♦s p✲s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s ❞❡ X✳
✶✻ ❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r
❊①❡♠♣❧♦ ✹✳ ❖s 0✲s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠♦ ♣♦♥t♦s ❞❡ X✱ ♦s 1✲
s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s ♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞♦s ❝♦♠♦ ❝❛♠✐♥❤♦s ❞❡ X ❡ ♦s 2✲s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s
♣♦❞❡♠ s❡r ✐❞❡♥t✐✜❝❛❞♦s ❡♠ X✱ ❝♦♠♦ ♥❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳
ix =10
ix =10
If(x )0
X
If
x0
Ix2
ix1
x0
If(x )0 X
If
If(x )1
If(x )2
x0
ix1 If(x )
0 X
If
If(x )1
❉❡✜♥✐çã♦ ✶✼✳ P❛r❛ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦X ❡ ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦R✱ ♦R✲♠ó❞✉❧♦ ❞❛sp✲❝❛❞❡✐❛s s✐♥❣✉❧❛r❡s ❡♠ X✱ Sp(X)✱ é ♦ ♠ó❞✉❧♦ ❧✐✈r❡ s♦❜r❡ R ♥♦ q✉❛❧ ❛ ❜❛s❡ B é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s
p✲s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s ❞❡ X✱ ✐st♦ é✱ B =Cp(X) ={φ : σp → X;φ é ✉♠ p✲s✐♠♣❧❡①♦ s✐♥❣✉❧❛r}✱
❝✉❥❛ ♥♦t❛çã♦ é Sp(X) = hCp(X)i✳
❖❜s❡r✈❛çã♦ ✺✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ Sp(X) ♥ã♦ sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞❡ σp ❡♠ X✱ sã♦ ❝♦♠❜✐✲
♥❛çõ❡s ❧✐♥❡❛r❡s ❢♦r♠❛✐s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥♦ ❛♥❡❧ R✱ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s φ : σp → X. ❯♠
❡❧❡♠❡♥t♦ ❞❡ Sp(X) t❡♠ ❛ ❢♦r♠❛
X
φ
nφφ✱ s❡♥❞♦ nφ ✐♥t❡✐r♦s ❡ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡❧❡s
♥ã♦ ♥✉❧♦s✳
✸✳✷ ❖♣❡r❛❞♦r ❜♦r❞♦
❉❡✜♥✐çã♦ ✶✽✳ ❙❡ φ é ✉♠ p✲s✐♠♣❧❡①♦ s✐♥❣✉❧❛r ❡♠ X ✭φ : σp → X✮ ❡ i é ✉♠ ✐♥t❡✐r♦ ❝♦♠
0≤i≤p✱ ❞❡✜♥✐♠♦s ❛ i✲és✐♠❛ ❢❛❝❡ ❞❡ φ ♣♦r Fi(φ)✱ ✉♠ (p−1)✲s✐♠♣❧❡①♦ s✐♥❣✉❧❛r ❡♠ X✱ ♣♦r
Fi(φ)(t0,· · · , tp−1) =φ(t0, t1,· · · , ti−1,0, ti,· · · , tp−1).
❊①❡♠♣❧♦ ✺✳ ◆❛ ✜❣✉r❛ ❛ s❡❣✉✐r✱ ✐❧✉str❛♠♦s ♦ ❝❛s♦ p= 2 ❡ i= 2✳
x0 Ix2
ix1
If(x )0 X
If
If(x )1
If(x )2
x0 ix1
❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r ✶✼
❉❡✜♥✐çã♦ ✶✾✳ ❖ ♦♣❡r❛❞♦r ❜♦r❞♦ ∂p :Sp(X)→Sp−1(X) é ❞❡✜♥✐❞♦ ♣♦r
∂p =F0−F1+F2− · · ·+ (−1)nFn=
X
(−1)iFi.
Pr♦♣♦s✐çã♦ ✽✳ ❆ ❝♦♠♣♦s✐çã♦ ∂p−1 ◦∂p ❡♠
· · · →Sp(X) ∂p
→Sp−1(X)
∂p−1
→ Sp−2(X)→ · · ·
s❡ ❛♥✉❧❛ ♣❛r❛ t♦❞♦ p✱ ♦✉ s❡❥❛✱ ∂p−1◦∂p = 0✳
❉❡♠♦♥str❛çã♦✿ ❬✺❪✱ Pr♦♣♦s✐çã♦ ✶✳✸✳
❉❡✈✐❞♦ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ s❡X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ♦ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s s✐♥❣✉❧❛r ❞❡ X s♦❜r❡ ♦ ❛♥❡❧ R é {Sp(X)}p≥0 ❝♦♠ ❛♣❧✐❝❛çõ❡s ∂p :Sp(X)→ Sp−1(X) ✭s❡ p = 0 ✱
❝♦♥s✐❞❡r❡ ∂p = 0✮ ✿
· · ·∂p+2
→ Sp+1(X)
∂p+1
→ Sp(X) ∂p
−→Sp−1(X)
∂p−1
→ · · · −→S1(X)
∂1
→S0(X)
∂0
→0.
❉❡✜♥✐çã♦ ✷✵✳ ❯♠ ❡❧❡♠❡♥t♦ c ∈ Sp(X) é ✉♠ p✲❝✐❝❧♦ s❡ ∂p(c) = 0 ❡ ✉♠ ❡❧❡♠❡♥t♦
d ∈ Sp(X) é ✉♠ p✲❜♦r❞♦ s❡ ❡①✐st❡ e ∈ Sp+1(X) t❛❧ q✉❡ d = ∂p+1(e)✳ ❆ss✐♠✱ Zp(X) =
=Ker ∂p ={c∈Sp(X)|c é ✉♠ p✲❝✐❝❧♦} ❡ Bp(X) =Im ∂p+1 ={d∈Sp(X)|d é ✉♠ p✲❜♦r❞♦}.
❉❡✜♥✐çã♦ ✷✶✳ ❙❡❥❛ X ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✳ ❈♦♥s✐❞❡r❡ Sp(X)✱ ∂p :Sp(X)→Sp−1(X)✱ Zp(X)
❡ Bp(X)✳ ❖ p✲és✐♠♦ ❣r✉♣♦ ❞❡ ❤♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ❞❡ X✱ é
Hp(X) =
Ker ∂p
Im ∂p+1
= Zp(X)
Bp(X)
=
c=c+Bp(C)
c∈Zp(X) .
✸✳✸ ❆ ❤♦♠♦❧♦❣✐❛ ❞♦ ♣♦♥t♦
❙❡❥❛X ={a}❡R✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✳ ❖ ❝♦♥❥✉♥t♦ ❞♦sn✲s✐♠♣❧❡①♦s s✐♥❣✉❧❛r❡s ❞❡ X é ❞❛❞♦ ♣♦r Cp(X) = {φ : σp → X;φ é ❝♦♥tí♥✉❛}✱ ♦✉ s❡❥❛✱ ♣❛r❛ ❝❛❞❛ p ≥ 0 ❡①✐st❡ ✉♠
ú♥✐❝♦ p✲s✐♠♣❧❡①♦ s✐♥❣✉❧❛r ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r φp✱ ✐st♦ é✱ Cp(X) = {φp}✳ ▲♦❣♦✱ Sp(X) ≃ R✱
♣❛r❛ t♦❞♦n✳
❖❜s❡r✈❡ q✉❡ s❡ p ≥ 1✱ ❛ ✐✲és✐♠❛ ❢❛❝❡ ❞❡ φp é ❞❛❞❛ ♣♦r Fi(φp) = φp−1✱ ♣❛r❛ t♦❞♦ i✱ ♣♦✐s
Fi(φi)∈Cp−1(X) = {φp−1}✳ ❖ ♦♣❡r❛❞♦r ❜♦r❞♦ ∂p :Cp(X)→Cp−1(X)é ❞❡✜♥✐❞♦ ♣♦r
∂p(φp) =
X
(−1)iFiφp =
X
(−1)iφp−1 =
(
0, s❡ ♣ é í♠♣❛r❀ φp−1, s❡ ♣ é ♣❛r✳
▲♦❣♦✱ s❡ p ❢♦r í♠♣❛r✱ ❡♥tã♦ ∂p : Cp(X) →Cp−1(X) é ♦ ❤♦♠♦♠♦r✜s♠♦ ♥✉❧♦ ❡ s❡ p ❢♦r ♣❛r✱
❡♥tã♦ ∂p :Cp(X)→Cp−1(X)é ✉♠ ✐s♦♠♦r✜s♠♦✳
◆♦t❡ q✉❡ ∂0 é ♦ ❤♦♠♦♠♦r✜s♠♦ ♥✉❧♦✳ ❆ss✐♠✱
H0(X) =
Ker ∂0
Im ∂1
= S0(X)
{0} ≃R.
✶✽ ❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r
❙❡ p >0 ❡ pé í♠♣❛r✱ ❡♥tã♦ p+ 1 é ♣❛r✱ ❡♥tã♦
Hp(X) =
Ker ∂p
Im ∂p+1
= Sp(X)
Sp(X)
≃ {0}.
❙❡ p >0 ❡ pé ♣❛r✱ ❡♥tã♦ p+ 1 é í♠♣❛r✱ ❡♥tã♦
Hp(X) =
Ker ∂p
Im ∂p+1
= {0}
{0} ≃ {0}.
P♦rt❛♥t♦✱
Hp(X) =
(
R, s❡p= 0 0, s❡p > 0✳
✸✳✹ ❍♦♠♦❧♦❣✐❛ s✐♥❣✉❧❛r ♥♦ ♥í✈❡❧ ③❡r♦
Pr♦♣♦s✐çã♦ ✾✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✱ ♥ã♦ ✈❛③✐♦✱ ❡♥tã♦ H0(X) ≃ R✱ ♦♥❞❡
R é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡✳
❉❡♠♦♥st❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛s s✐♥❣✉❧❛r
· · · −→S1(X)
∂1
→S0(X)
∂0
→0.
❚❡♠♦s q✉❡ Ker ∂0 =S0(X)❡ ❛ss✐♠ H0(X)≃
S0(X)
B0(X)
✳ ❱❛♠♦s ♠♦str❛r q✉❡ S0(X)
B0(X) ≃R✳
❈♦♥s✐❞❡r❡ ♦ R✲❤♦♠♦♠♦r✜s♠♦ f :S0(X)→R ❞❡✜♥✐❞❛ ♣♦r f
X
φ
nφφ
!
=X
φ
nφ✳
◆♦t❡ q✉❡ f é s♦❜r❡❥❡t♦r❛✱ ♣♦✐s ❞❛❞♦n ∈R✱ ❡s❝♦❧❤❛ φ∈C0(X) ❡♥tã♦ f(nφ) =n✳
❙❡❥❛d∈B0(X) = Im ∂1✱ ❡①✐st❡ ✉♠e=n1φ1+n2φ2+· · ·+nrφr ∈S1(X)t❛❧ q✉❡∂1(e) = d✱
♦♥❞❡ ni ∈ R ❡ φi : σ1 = [x0, x1] → X sã♦ ❝❛♠✐♥❤♦s ❡♠ X✳ ❙❡❥❛♠ ai = φi(x0) ❡ bi = φi(x1)✱
♣❛r❛ i= 1,2,· · · , r✳ ◆♦t❡ q✉❡ ∂1(φi) =bi−ai✳ ❊♥tã♦
d=∂1(n1φ1+n2φ2+· · ·+nrφr) =n1∂1(φ1) +n2∂1(φ2) +· · ·+nr∂1(φr)
=n1(b1−a1) +n2(b2−a2) +· · ·+nr(br−ar)
=n1b1−n1a1+n2b2−n2a2+· · ·+nrbr−nrar.
❆ss✐♠✱
f(d) = f(n1b1−n1a1+n2b2−n2a2+· · ·+nrbr−nrar)
= (n1−n1) + (n2−n2) +· · ·+ (nr−nr) = 0
❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r ✶✾
P♦rt❛♥t♦✱ d∈Kerf✱ ♦✉ s❡❥❛✱ B0(X)⊂Kerf✳
❆❣♦r❛✱ s❡❥❛ y ∈Kerf✳ ❉❡ss❡ ♠♦❞♦✱ y=n1φ1+n2φ2+· · ·+nrφr ∈S0(X)✱ t❛❧ q✉❡ f(y) =
n1+n2+· · ·+nr = 0✳ ◆♦t❡ q✉❡ φi :σ0 ={x0}={1} →X sã♦ ♣♦♥t♦s ❡♠ X✳ P❛r❛ ❢❛❝✐❧✐t❛r✱
❞❡♥♦t❡♠♦s φi(1) =wi✳ ❉❡ss❡ ♠♦❞♦✱ y=n1φ1+n2φ2+· · ·+nrφr =n1w1+n2w2+· · ·+nrwr✳
❋✐①❡ ✉♠ ♣♦♥t♦ a∈X ❡✱ ❝♦♠♦ ♣♦r ❤✐♣ót❡s❡✱X é ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✱ s❡❥❛♠φi : [x0, x1]→X
❝❛♠✐♥❤♦s ✉♥✐♥❞♦ ❛ ❛♦s ♣♦♥t♦swi✱ ✐st♦ é✱φi(x0) =a ❡ φi(x1) = wi✱ ♣❛r❛ t♦❞♦ i= 1,· · · , r✳
❈♦♥s✐❞❡r❡ e =n1φ1+n2φ2+· · ·+nrφr ∈S1(X) ❡ ♠♦str❡♠♦s q✉❡ ∂1(e) =y✳
❉❡ ❢❛t♦✱
∂1(e) =∂1(n1φ1+n2φ2+· · ·+nrφr)
=n1∂1(φ1) +n2∂1(φ2) +· · ·+nr∂1(φr)
=n1(w1−a) +n2(w2−a) +· · ·+nr(wr−a)
=n1w1 +n2w2+· · ·+nrwr−(n1+n2+· · ·+nr)a
=n1w1 +n2w2+· · ·+nrwr−f(y)a
=n1w1 +n2w2+· · ·+nrwr−0.a
=n1w1 +n2w2+· · ·+nrwr =y
P♦rt❛♥t♦✱ y∈B0(X)✱ ♦✉ s❡❥❛✱ Kerf ⊂B0(X)✳ ▲♦❣♦✱ B0(X) =Kerf✳
❙❡❣✉❡ ❞♦ t❡♦r❡♠❛ ❞♦ ✐s♦♠♦r✜s♠♦ ❞❡ R✲♠ó❞✉❧♦s q✉❡
H0(X)≃
S0(X)
B0(X)
= S0(X)
kerf ≃R.
❊①❡♠♣❧♦ ✻✳ ❖ t♦r♦ é ✉♠ ❡s♣❛ç♦ ❝♦♥❡①♦ ♣♦r ❝❛♠✐♥❤♦s✱ ❧♦❣♦ H0(X)≃R✳
▲❡♠❛ ✷✳ Hk(PCα)≈PαHk(Cα).
❉❡♠♦♥st❛çã♦✿ ◆♦t❡ q✉❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♠♣❧❡①♦ ❞❡ ❝❛❞❡✐❛sP
Cα t❡♠♦s
Zk
X
Cα=X(Zk(Cα)) ❡ Bk
X
Cα=X(Bk(Cα))
❆ss✐♠ s❡♥❞♦
✷✵ ❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r
Hk
X
Cα = Zk
X
Cα/Bk
X
Cα
= X(Zk(Cα))/
X
(Bk(Cα))
≈ X(Zk(Cα)/Bk(Cα))
= XHk(Cα).
Pr♦♣♦s✐çã♦ ✶✵✳ ❙❡ X é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡ {Xα : α ∈ A} sã♦ ❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ♣♦r
❝❛♠✐♥❤♦s ❞❡ X✱ ❡♥tã♦
Hk(X)≈
X
α∈A
Hk(Xα).
❉❡♠♦♥str❛çã♦✿ ❊①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ♥❛t✉r❛❧ ψ :X
α∈A
Sk(Xα)→Sk(X) ❞❛❞♦ ♣♦r
ψ
X
φα
nφ,α.φα
!
:α∈A
!
=X
α∈A
X
φα
nφ,α.φα
!
❈♦♠♦ ♦s ❣r✉♣♦s ❡♥✈♦❧✈✐❞♦s sã♦ ❛❜❡❧✐❛♥♦s ❧✐✈r❡s✱ ψ ❞❡✈❡ s❡r ✉♠ ♠♦♥♦♠♦r✜s♠♦✳ ▼♦str❡♠♦s q✉❡ ψ é t❛♠❜é♠ ✉♠ ❡♣✐♠♦r✜s♠♦✳
❉❡ ❢❛t♦✱ s❡❥❛ φ∈Sk(X)✱ ♦❜s❡r✈❡ ♣r✐♠❡✐r♦ q✉❡
φ:σk →X
é ✉♠ k✲s✐♠♣❧❡①♦ s✐♥❣✉❧❛r✱ ❡♥tã♦ φ(σk) ❡stá ❝♦♥t✐❞♦ ❡♠ ❛❧❣✉♠ Xα✱ ♣♦rq✉❡ σk é ❝♦♥❡①♦ ♣♦r
❝❛♠✐♥❤♦s ❡ φé ❝♦♥tí♥✉❛✳ ❆ss✐♠✱ ♣❛r❛ q✉❛❧q✉❡r ✉♠ ❞❡ss❡s φé ❛ss♦❝✐❛❞♦ ✉♠ ú♥✐❝♦φα ∈Sk(Xα)
❝♦♠ ψ(φα) =φ. P♦rt❛♥t♦✱ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ♣❛r❛ ❝❛❞❛ k.❆❧é♠ ❞✐ss♦✱ φ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡
❝❛❞❡✐❛s ❡♥tr❡ ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s✱ ❞❡ ♠♦❞♦ q✉❡
Hk(X)≈Hk
X
α∈A
S∗(Xα)
!
.
❋✐♥❛❧♠❡♥t❡✱ s❡❣✉❡ ❞♦ ▲❡♠❛ ✷ q✉❡
Hk
X
α∈A
S∗(Xα)
!
≈X
α∈A
Hk(Xα).
❊①❡♠♣❧♦ ✼✳ ❈♦♥s✐❞❡r❡ X ❝♦♠♦ s❡♥❞♦ ❛ ✉♥✐ã♦ ❞❡ ❞♦✐s t♦r♦s ❞✐s❥✉♥t♦s✱ ❡♥tã♦ s❡❣✉❡ ❞❛s ♣r♦♣♦✲ s✐çõ❡s ✾ ❡ ✶✵ q✉❡ H0(X)≃R⊕R✳
❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r ✷✶
✸✳✺ ❍♦♠♦♠♦r❢✐s♠♦ ■♥❞✉③✐❞♦ ❡♠ ❍♦♠♦❧♦❣✐❛
❙❡❥❛♠ ❳✱ ❨ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ f : X → Y ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❱❛♠♦s ❞❡✜♥✐r ✉♠ ❤♦♠♦♠♦r✜s♠♦ ✐♥❞✉③✐❞♦ ❞❡ ❢ ❡♠ ❤♦♠♦❧♦❣✐❛✱ ✐st♦ é✱ f∗ :Hn(X)−→Hn(Y)✳
Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡✜♥✐r❡♠♦s ✉♠ ❤♦♠♦♠♦r✜s♠♦ f# ❡♥tr❡ ♦s ❣r✉♣♦s ❞❡ ♥✲❝❛❞❡✐❛s Sn(X) ❡
Sn(Y)✳ ❈♦♠♦ Sn(X) ❡ Sn(Y) sã♦ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ❧✐✈r❡s ❜❛st❛rá ❞❡✜♥✐r ❛ ❛♣❧✐❝❛çã♦ ♥❛ ❜❛s❡
Cn(X) ❞❡Sn(X) ❡ ❡♥tã♦ ❡st❡♥❞❡r ♣♦r ❧✐♥❡❛r✐❞❛❞❡✳
❉❡✜♥✐çã♦ ✷✷✳ ❉❛❞❛ f :X →Y ❝♦♥tí♥✉❛✱ ❞❡✜♥✐♠♦s
f# :Sn(X) = hCn(X)i −→Sn(Y)
♥♦s ❣❡r❛❞♦r❡s✱ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
φ∈Cn(X)⇒f#(φ) =f ◦φ:σn→Y
(f◦φ ∈Cn(Y)⊂Sn(Y), ♥ ✲ s✐♠♣❧❡①♦ s✐♥❣✉❧❛r ❡♠ ❨ )
❆ ❡①t❡♥sã♦ ♥❛t✉r❛❧ ♣♦r ❧✐♥❡❛r✐❞❛❞❡ ♣❛r❛ Sn(X) é ❞❛❞❛ ♣♦r
f#
X
nφφ
=Xnφ(f◦φ).
x0
Ix2
ix1
If(x )0 IX
If
If(x )1
If(x )2 If
IY
I(fof)(x )0
I(fof)(x )1
I(fof)(x )2
If of
❖❜s❡r✈❛çã♦ ✻✳ {f#}n é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❝❛❞❡✐❛ ❡♥tr❡ ♦s ❝♦♠♣❧❡①♦s ❞❡ ❝❛❞❡✐❛s {Sn(X)} ❡
{Sn(Y)}✱ ✐st♦ é✱ f#◦∂ =∂◦f#.
Pr♦♣♦s✐çã♦ ✶✶✳ i)f#(Zn(X))⊂Zn(Y)
ii)f#(Bn(X))⊂Bn(Y)
✷✷ ❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r
❉❡♠♦♥str❛çã♦✿ ■✮ ❙❡❥❛
x∈f#(Zn(X)) =⇒x=f#(c), c∈Zn(X) =⇒∂(c) = 0.
❊♥tã♦✱
∂(x) = ∂(f#(c)) =f#(∂(c)) =f#(0) = 0∈Zn(Y).
■■✮ ❙❡❥❛
x∈f#(Bn(X)) =⇒x=f#(c), c∈Bn(X).
❊♥tã♦✱ ❡①✐st❡ d∈Sn+1(X) t❛❧ q✉❡ c=∂n+1(d)✳ ▲♦❣♦✱
x=f#◦∂n+1(d) = ∂n+1◦f#(d) =⇒x∈Bn(Y),
♣♦✐s f#(d)∈Sn+1(Y)✳
❈♦r♦❧ár✐♦ ✷✳ ❆ ❛♣❧✐❝❛çã♦ f# : Sn(X) → Sn(Y) ✐♥❞✉③ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❜❡♠ ❞❡✜♥✐❞♦
f :H∗(X)→H∗(Y)✱ ✐st♦ é✱ fn:Hn(X)→Hn(Y), ♣❛r❛ t♦❞♦ n✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛
w=Xnφφ+Bn(X)∈Hn(X) =
Zn(X)
Bn(X)
=⇒
=⇒f∗(w) = f#
X
nφφ
+Bn(Y) =
X
nφf#(φ) +Bn(Y) =
=Xnφ(f◦φ) +Bn(Y)∈Hn(Y) =
Zn(Y)
Bn(Y)
.
✸✳✻ ■♥✈❛r✐â♥❝✐❛ t♦♣♦❧ó❣✐❝❛ ❡ ❤♦♠♦tó♣✐❝❛
❚❡♦r❡♠❛ ✷✳ ❙❡ ❳ é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ ❞♦ ❘n ✭❡♠ ♣❛rt✐❝✉❧❛r s❡ X = ❘n✮✱ ❡♥tã♦
Hp(X) = 0 ♣❛r❛ t♦❞♦ p≥1✳
❉❡♠♦♥str❛çã♦✿ ❬✺❪✱ ✶✳✽✳
❈♦r♦❧ár✐♦ ✸✳ Hp(❘n) =
(
R, s❡ ♣ ❂ ✵❀
0, ❝❛s♦ ❝♦♥trár✐♦ ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ❘n é ❝♦♥❡①♦✱ s❡❣✉❡ q✉❡ H
0(❘n) ≃ R✳ ❆❧é♠ ❞✐ss♦✱ Hj(❘n) = 0 ♣❛r❛
t♦❞♦ j ≥1✱ ✉♠❛ ✈❡③ q✉❡ ❘n é ❝♦♥✈❡①♦✳
❖❜s❡r✈❛çã♦ ✼✳ ■✮ ❙❡ Id: X → X ❞❡♥♦t❛ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❳✱ ❡♥tã♦ Id∗ : Hn(X)→ Hn(X) é
♦ ❤♦♠♦♠♦r✜s♠♦ ✐❞❡♥t✐❞❛❞❡✱ ✐st♦ é✱ Id∗ =IdH(X)✳
■■✮ ❙❡ f :X →Y ❡ g :Y →Z sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♥tã♦✿ (g◦f)∗ =g∗◦f∗.
❆ ❞❡❢✐♥✐çã♦ ❞❛ ❍♦♠♦❧♦❣✐❛ ❙✐♥❣✉❧❛r ✷✸
❚❡♦r❡♠❛ ✸✳ ✭■♥✈❛r✐â♥❝✐❛ t♦♣♦❧ó❣✐❝❛✮✿ ❙❡ f : X → Y é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡♥tã♦ f∗ :Hn(X)→Hn(Y) é ✉♠ ✐s♦♠♦r✜s♠♦ ♣❛r❛ t♦❞♦ ♥✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ ❢ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❡♥tã♦ ❢ é ❜✐❥❡t✐✈❛✱ ❝♦♥tí♥✉❛ ❡ ♣♦ss✉✐ ✐♥✈❡rs❛ g = f−1 : Y →X t❛♠❜é♠ ❝♦♥tí♥✉❛✳ ❆ss✐♠g ◦f =Id
X ❡ f ◦g =IdY✳ ❙❛❜❡♠♦s q✉❡ f∗ é ✉♠
❤♦♠♦♠♦r✜s♠♦ ❡ (g◦f)∗ =g∗ ◦f∗ ❡ Id∗ = IdH(X)✳ ▲♦❣♦✱ f∗◦g∗ = (f ◦g)∗ = Id∗ =IdH(Y) ❡
g∗ ◦f∗ = (g ◦f)∗ = Id∗ = IdH(X)✱ ✐ss♦ ♠♦str❛ q✉❡ f∗ é ❜✐❥❡t♦r❛ ❡ g∗ é ❛ ✐♥✈❡rs❛ ❞❡ f∗✱ ✐st♦ é✱
(f∗)−1 =g∗ = (f−1)∗. ❆ss✐♠✱ f∗ é ✉♠ ✐s♦♠♦r✜s♠♦✳ ❉❡✜♥✐çã♦ ✷✸✳ ❙❡❥❛ ■ ♦ ✐♥t❡r✈❛❧♦ ❬✵✱✶❪ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ ❡ s❡❥❛♠ ❳✱ ❨ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ❡ f, g : X → Y ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✳ ❉✐③❡♠♦s q✉❡ ❢ ❡ ❣ sã♦ ❤♦♠♦tó♣✐❝❛s ❡ ❞❡♥♦t❛♠♦s f ∼ g✱ s❡
❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ F :X×I →Y, t❛❧ q✉❡ F(x,0) =f(x) ❡ F(x,1) = g(x), ♣❛r❛ t♦❞♦ x∈X.
❖❜s❡r✈❛çã♦ ✽✳ ❆ r❡❧❛çã♦ ❞❡ ❤♦♠♦t♦♣✐❛ (∼) é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛
❚❡♦r❡♠❛ ✹✳ ❙❡ f, g :X →Y sã♦ ❤♦♠♦tó♣✐❝❛s✱ ❡♥tã♦ f∗ =g∗ :Hn(X)→Hn(Y), ♣❛r❛ t♦❞♦ n.
❉❡♠♦♥tr❛çã♦✿ ❬✺❪✱ ✶✳✶✵✳
❉❡✜♥✐çã♦ ✷✹✳ ❙❡❥❛♠ f : X → Y ❡ g : Y → X ❛♣❧✐❝❛çõ❡s ❡♠ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❙❡ ❛s ❝♦♠♣♦s✐çõ❡s f ◦g ❡ g◦f sã♦ ❝❛❞❛ ✉♠❛ ❤♦♠♦tó♣✐❝❛ ❛ s✉❛ r❡s♣❡❝t✐✈❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡
(f◦g ∼IdY, g◦f ∼IdX)✱ ❡♥tã♦ ❢ ❡ ❣ sã♦ ❞✐t❛s ✐♥✈❡rs❛s ❤♦♠♦tó♣✐❝❛s ✉♠❛ ❞❛ ♦✉tr❛✳ ❯♠❛ ❛♣❧✐✲
❝❛çã♦ f :X →Y é ❝❤❛♠❛❞❛ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❤♦♠♦t♦♣✐❛ s❡ ❢ t❡♠ ✉♠❛ ✐♥✈❡rs❛ ❤♦♠♦tó♣✐❝❛✱ ✐st♦ é✱ ❡①✐st❡ g :Y →X t❛❧ q✉❡ f◦g ∼IdY ❡ g◦f ∼IdX.
❊①❡♠♣❧♦ ✽✳ ❙❡ ❢ é ❤♦♠❡♦♠♦r✜s♠♦✱ ❡♥tã♦ ❢ é ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❤♦♠♦t♦♣✐❛✳
❉❡ ❢❛t♦✱ s❡♥❞♦ ❢ ✉♠ ❤♦♠❡♦♠♦r✜s♠♦✱ ❡①✐st❡ g = f−1. ▲♦❣♦✱ f ◦ g = Id
Y ∼ IdY ❡
g◦f =IdX ∼IdX.
❉❡✜♥✐çã♦ ✷✺✳ ❙❡❥❛♠ ❳ ❡ ❨ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s✳ ❉✐③❡♠♦s q✉❡ ❳ ❡ ❨ t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ s❡ ❡①✐st❡ f :X →Y ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❤♦♠♦t♦♣✐❛✳
❊①❡♠♣❧♦ ✾✳ ❆ ❝✐r❝✉♥❢❡rê♥❝✐❛ S1 t❡♠ ♦ ♠❡s♠♦ t✐♣♦ ❞❡ ❤♦♠♦t♦♣✐❛ ❞❡ ❘2\{0}.
❈♦♥s✐❞❡r❡ ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ f : S1 → ❘2\{0} t❛❧ q✉❡ f(u) = u ❡ g : ❘2\{0} → S1
t❛❧ q✉❡ g(u) = u
kuk✳ ❊♥tã♦ t❡♠♦s q✉❡ (g ◦ f)(u) = g(f(u)) =
u
kuk = u = idS1 ❡
(f◦g)(u) =f(g(u)) = u
kuk✳ ❈♦♠♦
h
u, u
kuk
i
⊂❘2\{0}✱ ♣❛r❛ t♦❞♦u∈❘2\{0}✱ ♣♦✐s 1
kuk 6= 0✱ ❡♥tã♦ ❞❡✜♥✐♠♦s H :❘2\{0} ×I →❘2\{0} ♣♦r H(x, t) = (1−t) u
kuk +tu✳ ❊♥tã♦✱ f◦g ∼id❘2\{0}✳ ❚❡♦r❡♠❛ ✺✳ ❙❡ f : X →Y é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❤♦♠♦t♦♣✐❛✱ ❡♥tã♦ f∗ : Hn(X)→ Hn(Y) é
✉♠ ✐s♦♠♦r✜s♠♦✱ ♣❛r❛ ❝❛❞❛ ♥✳
❉❡♠♦♥str❛çã♦✿ ❙❡♥❞♦ ❢ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❤♦♠♦t♦♣✐❛✱ ❡①✐st❡ ✉♠❛ ✐♥✈❡rs❛ ❤♦♠♦tó♣✐❝❛ ❣ ❞❡ ❢✳ ▲♦❣♦✱ f ◦g ∼IdY ❡g◦f ∼IdX.
P❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ (f ◦g)∗ =Id∗ ❡ (g◦f)∗ =Id∗. ❆ss✐♠✱ f∗◦g∗ =Id ❡g∗◦f∗ =Id, ♦✉ s❡❥❛✱ g∗ =f∗−1.
P♦rt❛♥t♦ ✱ f∗ é ✉♠ ✐s♦♠♦r✜s♠♦✳
