Gomotopiya v matematici ponyattya algebrichnoyi topologiyi sho formalizuye ponyattya neperervnoyi deformaciyi odnogo ob yekta v inshij Za dopomogoyu gomotopiyi viznachayutsya gomotopichni grupi sho ye vazhlivimi invariantami v algebrichnij topologiyi Zmist 1 Formalne viznachennya 2 Pov yazani viznachennya 3 Gomotopichna ekvivalentnist 4 Gomotopichna grupa 5 Vlastivosti 6 Prikladi 7 Posilannya 8 LiteraturaFormalne viznachennya RedaguvatiNehaj X displaystyle X nbsp ta Y displaystyle Y nbsp topologichni prostori i f ta g dva neperervnih vidobrazhennya z prostoru X displaystyle X nbsp v prostir Y displaystyle Y nbsp Todi vidobrazhennya f nazivayetsya gomotopnim vidobrazhennyu g yaksho isnuye neperervne vidobrazhennya H X 0 1 Y displaystyle H colon X times 0 1 to Y nbsp take sho f x H x 0 displaystyle f x H x 0 nbsp i g x H x 1 displaystyle g x H x 1 nbsp dlya x X Dane neperervne vidobrazhennya nazivayetsya gomotopiyeyu Pov yazani viznachennya Redaguvati nbsp Gomotopichna ekvivalentnist bublika i chashkiGomotopichnij invariant ce harakteristika prostoru yaka zberigayetsya pri gomotopichnij ekvivalentnosti topologichnih prostoriv Tobto yaksho dva prostori gomotopno ekvivalenti to voni mayut odnakovu harakteristiku Napriklad zv yaznist fundamentalna grupa ejlerova harakteristika Yaksho na deyakij pidmnozhini A X F t a f a displaystyle A subset X F t a f a nbsp dlya vsih t displaystyle t nbsp pri a A displaystyle a in A nbsp to F displaystyle F nbsp nazivayetsya gomotopiyeyu vidnosno A displaystyle A nbsp a f displaystyle f nbsp i g displaystyle g nbsp gomotopnimi vidnosno A displaystyle A nbsp Izotopiya gomotopiya topologichnogo prostoru X displaystyle X nbsp po topologichnomu prostoru Y displaystyle Y nbsp tobto f t X Y t 0 1 displaystyle f t colon X to Y t in 0 1 nbsp v yakij pri bud komu t displaystyle t nbsp vidobrazhennya f t displaystyle f t nbsp ye gomeomorfizmom X displaystyle X nbsp na f X Y displaystyle f X subset Y nbsp Gomotopichna ekvivalentnist RedaguvatiGomotopichna ekvivalentnist topologichnih prostoriv X displaystyle X nbsp i Y displaystyle Y nbsp para neperervnih vidobrazhen f X Y displaystyle f colon X to Y nbsp i g Y X displaystyle g colon Y to X nbsp taka sho f g id Y displaystyle f circ g sim operatorname id Y nbsp i g f id X displaystyle g circ f sim operatorname id X nbsp tut displaystyle sim nbsp poznachaye gomotopichnu ekvivalentnist vidobrazhen V comu vipadku govoryat sho X displaystyle X nbsp i Y displaystyle Y nbsp gomotopno ekvivalentni abo X displaystyle X nbsp z Y displaystyle Y nbsp mayut odin gomotopnij tip Gomotopichna grupa RedaguvatiGomotopichna grupa prostoru PS p n PS ps 0 displaystyle Psi pi n Psi psi 0 nbsp ye grupoyu gomotopichnih klasiv neperervnih vidobrazhen f S n PS displaystyle f S n rightarrow Psi nbsp perevodyachi vidznachenu tochku sferi u tochku ps 0 displaystyle psi 0 nbsp iz dekotroyu operaciyeyu Sferu S n displaystyle S n nbsp mozhna neperervno j biyektivno vidobraziti u I n displaystyle I n nbsp de I 0 1 displaystyle I 0 1 nbsp Takim chinom gomotopichnu grupu mozhna viznachiti yak grupu gomotopichnih klasiv neperervnih vidobrazhen g I n PS displaystyle g I n rightarrow Psi nbsp yaki perevodyat granicyu u vidznachenu tochku g I n ps 0 displaystyle g partial I n psi 0 nbsp Operaciyu takih vidobrazhen mozhna viznachiti nastupnim chinom g 1 g 2 t 1 t n g 1 2 t 1 t 2 t n t 1 0 0 5 g 2 2 t 1 1 t 2 t n t 1 0 5 1 displaystyle g 1 g 2 t 1 t n begin cases g 1 2t 1 t 2 t n amp t 1 in 0 0 5 g 2 2t 1 1 t 2 t n amp t 1 in 0 5 1 end cases nbsp Vlastivosti RedaguvatiGomotopiya zadaye vidnoshennya ekvivalentnosti na mnozhini neperervnih vidobrazhen X Y displaystyle X to Y nbsp Refleksivnist Yaksho f X Y displaystyle f colon X to Y nbsp deyake neperervne vidobrazhennya todi funkciya H X I Y displaystyle H colon X times I to Y nbsp viznachena H x t f x displaystyle H x t f x nbsp bude gomotopiyeyu mizh f i f Simetrichnist Nehaj vidobrazhennya f X Y displaystyle f colon X to Y nbsp gomotopne vidobrazhennyu g X Y displaystyle g colon X to Y nbsp i H X I Y displaystyle H colon X times I to Y nbsp vidpovidna gomotopiya Todi g ye gomotopnim f z gomotopiyeyu H x t H x 1 t displaystyle H x t H x 1 t nbsp Tranzitivnist Nehaj vidobrazhennya f X Y displaystyle f colon X to Y nbsp gomotopne vidobrazhennyu g X Y displaystyle g colon X to Y nbsp i H X I Y displaystyle H colon X times I to Y nbsp vidpovidna gomotopiya Nehaj takozh vidobrazhennya g X Y displaystyle g colon X to Y nbsp gomotopne vidobrazhennyu h X Y displaystyle h colon X to Y nbsp i F X I Y displaystyle F colon X times I to Y nbsp vidpovidna gomotopiya Todi Todi f ye gomotopnim h z gomotopiyeyu dd G x t H x 2 t t 0 0 5 H x 2 t 1 t 0 5 1 displaystyle G x t begin cases H x 2t amp t in 0 0 5 H x 2t 1 amp t in 0 5 1 end cases nbsp dd dd Usi vidobrazhennya h t x H x t displaystyle h t x H x t nbsp ye neperervnimi Yaksho f f X Y g Y B h A X displaystyle f f colon X to Y g colon Y to B h colon A to X nbsp neperervni vidobrazhennya i H X I Y displaystyle H colon X times I to Y nbsp gomotopiya mizh f displaystyle f nbsp i f displaystyle f nbsp to g H h I displaystyle g circ H circ h times I nbsp ye gomotopiyeyu mizh g f h displaystyle g circ f circ h nbsp i g f h displaystyle g circ f circ h nbsp Prikladi RedaguvatiYaksho Y R m displaystyle Y mathbb R m nbsp to funkciyi f displaystyle f nbsp i g displaystyle g nbsp ye zavzhdi ye gomotopnimi Gomotopiya viznachayetsya H x t f x t g x f x displaystyle H x t f x t left g x f x right nbsp Mnozhini X 0 1 Y 0 1 displaystyle X 0 1 Y 0 1 nbsp ye ekvivalentnimi gomotopichno ale ne gomeomorfnimi Odinichne kolo S 1 displaystyle mathcal S 1 nbsp gomotopno ekvivalentne prostoru R 2 0 displaystyle mathbb R 2 setminus 0 nbsp M N p t lim N k f M f p t displaystyle M N pt cong lim rightarrow N k f M f pt nbsp de N k f displaystyle N k f nbsp aproksimuyuchi skinchenni modeli CW komplesu N displaystyle N nbsp Tut mi mayemo vidobrazhennya N k f M f p t N M f p t N M f p t k displaystyle N k f M f pt rightarrow N M f pt cong N M f pt forall k nbsp Otrimuyemo biyekciyu f lim N k f M f p t N M f displaystyle varphi lim rightarrow N k f M f pt rightarrow N M f nbsp Nehaj U G displaystyle U G nbsp gomotopichni prostori iz vidznachenoyu tochkoyu de G displaystyle G nbsp skinchenne j u nomu vikonuyetsya T 0 displaystyle T 0 nbsp Nehaj vidobrazhennya f g U G displaystyle f g U rightarrow G nbsp ye neperervnimi ta u U displaystyle forall u in U nbsp vikonuyetsya f x g x displaystyle f x in overline g x nbsp Todi voni ye gomotopnimi Dijsno mozhna pobuduvati gomotopiyu H U I G displaystyle H U times I rightarrow G nbsp iz nastupnimi vlastivostyami H u 0 f x displaystyle H u 0 f x nbsp H u t v u t 0 1 displaystyle H u t v u t in 0 1 nbsp Shob pokazati neperervnist vidobrazhezhennya H displaystyle H nbsp potribno pokazati sho H 1 p displaystyle H 1 bar p nbsp ye zamknenim dlya bud yakoyi tochki p f U v U displaystyle p in f U cup v U nbsp Yaksho v u p displaystyle v u in bar p nbsp to j f u p displaystyle f u in bar p nbsp Ce daye v 1 p 0 1 f 1 p 0 v 1 p 0 1 displaystyle v 1 bar p times 0 1 subseteq f 1 bar p times 0 cup v 1 bar p times 0 1 nbsp Todi H 1 p f 1 p 0 v 1 p 0 1 f 1 p 0 v 1 p 0 1 displaystyle H 1 bar p f 1 bar p times 0 cup v 1 bar p times 0 1 f 1 bar p times 0 cup v 1 bar p times 0 1 nbsp A vidtak vin ye zamknenim yak ob yednannya zamknenih mnozhin Posilannya RedaguvatiS Maksimenko Institut matematiki NAN Ukrayini Vstup do teoriyi gomotopij na YouTube Literatura RedaguvatiVasilev V A Vvedenie v topologiyu M FAZIS 1997 132 s ISBN 5 7036 0036 7 Rohlin V A Fuks D B Nachalnyj kurs topologii Geometricheskie glavy M Nauka 1977 Spener E Algebraicheskaya topologiya M Mir 1971 Otrimano z https uk wikipedia org w index php title Gomotopiya amp oldid 36730389
