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ÃÐÀÍÈ×ÍÛÉ ÎÏÅÐÀÒÎÐ
ÃÐÀÍÈ×ÍÛÉ ÎÏÅÐÀÒÎÐ: 1°. Ã. î.
êëåòî÷íîãî êîìïëåêñà — îïåðàòîð , îïðåäåëåííûé íà öåïÿõ êëåòî÷íîãî
êîìïëåêñà, ò. å. íà öåëî÷èñëåííûõ ôîðìàëüíûõ ëèíåéíûõ êîìáèíàöèÿõ
îðèåíòèðîâàííûõ êëåòîê êîìïëåêñà
, ãäå
— öåëûå ÷èñëà,
— îðèåíòèðîâàííûå êëåòêè. Íà
êàæäóþ êëåòêó
ðàçìåðíîñòè
îïåðàòîð
äåéñòâóåò
ïî ôîðìóëå
, (*)
ãäå
,
— âñå êëåòêè
ðàçìåðíîñòè
.
Ñèìâîë
îçíà÷àåò
öåëîå ÷èñëî (êîýôôèöèåíò èíöèäåíòíîñòè), îïðåäåëÿåìîå ñëåäóþùèì îáðàçîì. Ïóñòü
—
îðèåíòèðîâàííûå êëåòêè è
,
— ñîîòâåòñòâóþùèå èì õàðàêòåðèñòè÷åñêèå
îòîáðàæåíèÿ. Ãðàíèöà
ìíîæåñòâà
îòîáðàæàåòñÿ
ïîä äåéñòâèåì
â
-ìåðíûé îñòîâ
ïðîñòðàíñòâà
. Ýòîò îñòîâ
ñòàíäàðòíûì îáðàçîì
îòîáðàæàåòñÿ â ïðîñòðàíñòâî
;
(çäåñü
—
-ìåðíûé îñòîâ, à
îçíà÷àåò òîïîëîãè÷åñêîå
ïðîñòðàíñòâî
, â
êîòîðîì ïîäìíîæåñòâî
îáúÿâëåíî îäíîé òî÷êîé (òîïîëîãèÿ
îïðåäåëÿåòñÿ òîïîëîãèåé
). Ïðîñòðàíñòâî
èìååò êëåòî÷íîå ðàçáèåíèå,
ÿâëÿþùååñÿ áóêåòîì
-ìåðíûõ
ñôåð. Ðàññìîòðèì ñêâîçíîå îòîáðàæåíèå:
.
Âñÿêîå îòîáðàæåíèå îðèåíòèðîâàííîé ñôåðû
â áóêåò
îðèåíòèðîâàííûõ ñôåð
ãîìîòîïè÷åñêè
ýêâèâàëåíòíî (ñì. Ãîìîòîïè÷åñêàÿ ýêâèâàëåíòíîñòü) îòîáðàæåíèþ,
ïîñòðîåííîìó ïî ñëåäóþùåé êîíñòðóêöèè: ñíà÷àëà ñôåðà
ñòÿãèâàíèåì â òî÷êó íåñêîëüêèõ
ïîäìíîæåñòâ
,
, ïðåâðàùàåòñÿ â áóêåò
ñôåð
(Äëÿ ýòîãî íàäî
â êà÷åñòâå
âûáðàòü
ñôåðû ðàçìåðíîñòè
,
ïåðåñåêàþùèåñÿ ìåæäó ñîáîé òîëüêî â îäíîé òî÷êå.) Äàëåå áóêåò
îòîáðàæàåòñÿ â áóêåò
òàê: ñôåðû
áóêåòà
îòîáðàæàþòñÿ
â ñîîòâåòñòâóþùèå (ïî íîìåðó) ñôåðû áóêåòà
; ïðè ýòîì êàæäîå òàêñå îòîáðàæåíèå
,
(ñ òî÷íîñòüþ äî ãîìîòîïè÷åñêîé
ýêâèâàëåíòíîñòè) îïðåäåëÿåòñÿ öåëûì ÷èñëîì
— ñòåïåíüþ îòîáðàæåíèÿ.
Èòàê, ãðàíèöà øàðà
îòîáðàæåíà â
, à ñðåäè
åñòü òà ñôåðà, êîòîðàÿ
ÿâëÿåòñÿ îáðàçîì êëåòêè
ïðè
îòîáðàæåíèè
(ïîïîëíåííûì
îäíîé çàìûêàþùåé òî÷êîé).
Ïóñòü íîìåð ýòîé ñôåðû . Òîãäà
.
Äåéñòâèå îïåðàòîðà , îïðåäåëåííîå ôîðìóëîé (*),
ðàñïðîñòðàíÿåòñÿ íà ìíîæåñòâî âñåõ öåïåé ïî ëèíåéíîñòè, ò. å.
.
2°. Ã. î. ñèìïëèöèàëüíîãî êîìïëåêñà — îïåðàòîð , îïðåäåëåííûé íà öåïÿõ
ñèìïëèöèàëüíîãî êîìïëåêñà, ò. å. íà öåëî÷èñëåííûõ ëèíåéíûõ êîìáèíàöèÿõ
îðèåíòèðîâàííûõ ñèìïëåêñîâ
ãäå
— öåëûå ÷èñëà,
— îðèåíòèðîâàííûå ñèìïëåêñû.
Íà êàæäûé ñèìïëåêñ ðàçìåðíîñòè
îïåðàòîð
äåéñòâóåò ïî ôîðìóëå
, (**)
ãäå — ñîâîêóïíîñòü êîãåðåíòíî îðèåíòèðîâàííûõ
ãðàíåé ñèìïëåêñà
,
ò. å. åñëè
, òî
, çäåñü
âåðøèíû ñèìïëåêñà, à
— îðèåíòàöèÿ
ñèìïëåêñà.