Bott functions and the euler characteristic
In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds.
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| Datum: | 1999 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4743 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510907645296640 |
|---|---|
| author | Bondar', O. P. Бондарь, О. П. Бондарь, О. П. |
| author_facet | Bondar', O. P. Бондарь, О. П. Бондарь, О. П. |
| author_sort | Bondar', O. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds. |
| first_indexed | 2026-03-24T03:04:27Z |
| format | Article |
| fulltext |
Y}]K 515.164.174
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| I;OTTA H ~I'~JIEPOBA XAPAKTEPHCTHKA
In terms of the Eulerian characteristic, we obtain the condition for the existence of Bott functions on
differentiable manifolds with a set of critical points which consists of connected homeomorphic
submanifolds.
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ponoi xapaKTepHcTnKrl.
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Onpe~eJ1eHHe 1. ~ynr~u .~ f nasbtsaemc.~ ~ynr.uea Eomma [1], ecnu ~tno-
.~ecmeo ee Kpumuqecrux mo,wK nsn~emc~ necse.e,t~bt~t o6"beOunenue~t ne~btpo~raen-
nbtx znaaKux noa,~tHOeOOSpa3u~, ne nepeceKwou~uxc.~ c rpae;~t OM n.
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KUI, II~ - - TaK Ha3blBaeMlale Kpyr.rlue c1DyHKI_I;HH Mopca ~ pacCMOTpeH, HanpriHep, s
pa6oTax [2 - 7].
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~M n = ~ M n O ~+M n
f : ( M n, ~_M", O+M") ~ ( [0, 1 ] ,0, I ) ~ rna~Kaz qbynr..uriz.
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zooSpaau.~ ~ n c no~tou~bto npur.ae~tru neabtpo~Oennoft P-py~tru unDerca ~.,
ecru
M n = ~ n U g P x D ~ x D n-~'-p,
zOe g : P x OD ~ x D n -x -p --4 O+M n ~ z~aOtcoe a,ao~enue.
Onpe~tes[enne 3. Pasno~enue,~t ~tnoeooSpa3u.a M n Ha neabtpo~Oenntae P-
py~Ku na3o~e~t qbu~bmpat~uto
M 0 c M ! c . . . c M n = M n
maryto, ~mo O_M" x [0, 1 ] = M 0 u Ka~Ooe n-~tepnoe ,unoeoo6pa3ue M i no.ay~e-
no u3 M i_ l c no,~totqblo nputc.ae~tcu ne~btpo~ennbtx P-py~tex unOerca i. B c.ay-
qae, KozOa 3_M"= ~ , M 0 cocmoum us ue~btpo~gennbtX P-py~ter unOerca O.
PaCCMOTpHM Ha M n TaKHe qbynxaaVl BOTTa C KprlTrlqeCKlaMn nOZ~HOrOo6paan2-
MH P, cytI2eCTBOBaHHC KOT0pblX 3KBHBaJ-IeHTHO pa3dlO;KCnHIO M n Ha HCBI~IpO)K~eH-
n~e P-pyqKH. B c.qyqae P = S 1 Bce cl3yHKI~HH So- i ra $[BJ'IJ~tOTC~ TaKHMH (CM. [4, 8]).
Torzta cnpaseztnnao c~ezty~otaee yaaep:~t~eHHe.
YTaep~Ktlen~e. E c n u n a ~tnoeooSpa3uu M n c y t , e c m a y e m dpynr.t4u.~ l~omma
c KpumutwcKu~tu noO~tt~oeoo5pa3u,q~tu P, mo
n-,i-p
z ( M n) = z ( P ) 2 ( -1 )Xk~ , ,
x=o
zOe k~. ~ ~ucno P - p y ~ e r unOeKca ~ ~ p a s , w z c e n u u M n n a ne~btpoacOenn~e
P-py~Ku.
�9 O. H, SOHZLAPb, 1999
ISSN 0041-6053. Yrp. ~iam. ~.'vpn., 1999, m. 51, Na lO 1431
1 4 3 2 O . H . B O H ] I A P b
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, ~oga3ame ,~ , cmoo . BOCRO.rlIx.~BaBIIIHCb TeM, qTO 3 ~ J 1 e p o a a xapaKTCpHCTHKa
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pagTcpHCTh~ H ]la~ 2m)6og supe3aeMofl Tpaa]Ra cnpaee~Jmso COOTaomeHae
z(A 0 B) = ~(A)+x(B)- z(A n B),
no~md
x ( M n) = X ( M n ) + z ( P X D;~x D n - k - P ) - ; ~ ( P x aDXx D n-k 'P) =
= Z ( M n ) + z ( P ) - X ( P ) x ( S ~ - l ) = x ( M n) + z ( P ) [ 1 - Z ( S a ' - I ) ] =
= z (~" ) + (- 1)~-z(e).
Y~PIT~Ba,a TO, ,.i'ro ~t,TICpOS'-' xapazTcpecTaKa npoc"rpaiac'ra, o~eo H3 zo ' rop~x
.aS.5$IeTC,,q/IeE1)opMaI.II4OHHblM peTpaKTOM ~ p y r o r o , paBHU, no .nyqaeM r p e 6 y e M o e p a -
B(~HCTBO.
C.aeacmaue. Ec.nu 9~.aepooa xapagmepucmura Ko~ma~.mnozo ~t/-tozoo6pa3ua M n
5e3 rpa~ orrtnu~na om n)'.a~, mo 3.a.~ mozo, ttmoSbz Ma M n cyu4ecmoosa~a dps'nK~u~t
l~omma c rpumu~ecru~tu noO~tnozoo6pazu,~t~tu P, neo5xoOuMo, qmoSta a~,aepooa
x a p a r m e p u c m u r a ~tnozoo@azu,~ M n 5btna Kpamna a~t.nepooo~ xapato'nepucmu~e
noO~mozoo@azu,~ P.
1- lpa n > 4 I~ P = S l /~oKa3aHHoe yTsepxvJIeHrle o 6 p a T a o TcopeMe A3PIMoBa [2],
1. Bott R. "L,eeture on Morse theory; old and new//Bul l . Amcr. Math. Soe. - 1982. - 7, N ~ 2. - P. 331
- 358.
2. Asimov D. Round handles and non-singular Morsc - Smale f l o w s / / A n n . Math. - 1975. - 102,
N ~ 1 . - P . 4 1 - 5 4 .
3. Franks J. The periodic behavior of non-singular Morse - Smalc f l ows / /Comment , math. holy. -
1978. - 53. N ~ 2. - P. 279 - 294.
4. Miyosihi S. Foliated round surgery of codimension-one foliatcd manifo lds / /Topology. - 1983. -
21. N ~ 3. - P. 245 - 262.
5. Morgan .L Non-singular Morse - Smalc flows on 3-dimensional manifolds / / Ib id . - 1979. - 18,
N ~ 1 . - P . 41 - 5 3 .
6. Mamsee~ C. B.. Cl)o,~wnro A. T.. lllapro B. B. KpyrJn,~e qbyugRrta Mopca t4 a3o~uepreTa,4eczae
nonepxuocm mrrerpapyeM~ax ra~a.~=wronomax eaca'e~ // MaT. c6. - 1988. - 135, N ~ 3. - C. 325 -
345.
7. lllaptr B. B. Oyng~imt na Mnoroo6paaaax. - Kaea: Hay=<..,~y~ga, 1990. - 196 c.
8. Wall C. T. Formal deformations//Proe. London Math. S o c . - 1966. - 3, N ~ 2. - P. 342 - 352.
i'loJzyqcno 12.03.98
ISSN 0041-605/. Y~:p. ,uam. #,.'vpu.. 1999. m. 51. N ~ J O
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| id | umjimathkievua-article-4743 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:04:27Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ff/dbc9fa7f3cb5bf56e97c6cc9e964aaff.pdf |
| spelling | umjimathkievua-article-47432020-03-18T21:12:54Z Bott functions and the euler characteristic Функции Ботта и эйлерова характеристика Bondar', O. P. Бондарь, О. П. Бондарь, О. П. In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds. Одержано умову існування функцій Ботта на диферепційошіих многовидах, що мають множину критичних точок, яка складається із зв'язних гомеоморфних підмноговидів, у термінах ейлерової характеристики. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4743 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1431–1432 Український математичний журнал; Том 51 № 10 (1999); 1431–1432 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4743/6187 https://umj.imath.kiev.ua/index.php/umj/article/view/4743/6188 Copyright (c) 1999 Bondar' O. P. |
| spellingShingle | Bondar', O. P. Бондарь, О. П. Бондарь, О. П. Bott functions and the euler characteristic |
| title | Bott functions and the euler characteristic |
| title_alt | Функции Ботта и эйлерова характеристика |
| title_full | Bott functions and the euler characteristic |
| title_fullStr | Bott functions and the euler characteristic |
| title_full_unstemmed | Bott functions and the euler characteristic |
| title_short | Bott functions and the euler characteristic |
| title_sort | bott functions and the euler characteristic |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4743 |
| work_keys_str_mv | AT bondar039op bottfunctionsandtheeulercharacteristic AT bondarʹop bottfunctionsandtheeulercharacteristic AT bondarʹop bottfunctionsandtheeulercharacteristic AT bondar039op funkciibottaiéjlerovaharakteristika AT bondarʹop funkciibottaiéjlerovaharakteristika AT bondarʹop funkciibottaiéjlerovaharakteristika |