20.500.12556/RUL-140909 Poincaréjeva dualnost v de Rhamovi kohomologiji delo diplomskega seminarja Poincaré duality in de Rham cohomology V delu predstavimo koncept ovojnih števil in potencialnih polj iz analize ter razvijemo de Rhamovo teorijo, ki posploši tovrstna vprašanja in koncepte na gladke mnogoterosti. Osrednji objekti so kohomološki vektorski prostori, ki v posebnem primeru klasične vektorske analize na primer povedo, v kolikšni meri je na nekem poljubnem območju irotacijsko vektorsko polje tudi potencialno. Glavni rezultat je Poincaréjeva dualnost, ki na intuitivnem nivoju pravi, da je za sklenjeno orientabilno mnogoterost dimenzije $n$ dualni prostor kohomološkega prostora v dimenziji $k$ kar kohomološki prostor v dimenziji $n - k$. Začnemo s hitrim pregledom najpomemb- nejših osnov gladkih mnogoterosti in z definicijo diferencialnih form, ki so vir bogate algebraične strukture v središču de Rhamove teorije. Posplošimo Riemannov inte- gral na integriranje po gladkih mnogoterostih in dokažemo splošen Stokesov izrek. Razvijemo osnovno de Rhamovo teorijo, kjer vpeljemo Mayer-Vietorisovo zaporedje, ki je močno računsko orodje, hkrati pa pokažemo, da je teorija homotopsko inva- riantna. Na tej točki izračunamo kohomologijo evklidskega prostora in dokažemo Poincaréjevo lemo. Kot primer uporabe dokažemo Brouwerjev izrek o negibni točki in na koncu dokažemo Poincaréjevo dualnost. In this work we showcase the concept of a winding number and potential fields from analysis and we develop the de Rham theory, which generalizes problems and concepts of this kind to smooth manifolds. The central objects are the cohomology vector spaces, which tell us in the special case of the classical vector calculus, for example, how much is an irrotational vector field on any particular domain also a potential one. The main result is the Poincaré duality, which intuitively says that for a closed orientable $n$-dimensional manifold, the dual of a cohomology space in dimension $k$ is again a cohomology space in dimension $n - k$. We begin with a quick review of the essential basics of smooth manifolds and with the definition of differential forms, which are the source of the rich algebraic structure at the heart of de Rham theory. We generalize the Riemann integral to integration over smooth manifolds and we prove the general Stokes’ theorem. We develop the basics of de Rham theory, where we introduce the Mayer-Vietoris sequence, which is a powerful computational tool and we also show that the theory is homotopy invariant. It is here that we compute the cohomology of the euclidean space and prove the Poincaré lemma. Using these results we prove the Brouwer fixed point theorem and in the end we prove the Poincaré duality. de Rhamova kohomologija diferencialne forme mnogoterosti de Rham cohomology differential forms manifolds true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2022-09-21 08:15:06 2022-09-21 08:15:08 2024-05-29 12:00:01 0000-00-00 00:00:00 2022 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.1 127625 122597891 5504.pdf 5504.pdf 1 5B3BEBA9EDE32F51E113CF34C83FBAD8 f2582cc9b464ab4597acf4c7dc66dce4dbf268debbad43043d224a37383bb6fd ab47dab1-3974-11ed-92af-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=161753 Fakulteta za matematiko in fiziko 0 0 0