Euler Classes and Frobenius Algebras

Ranaivomanana, Valimbavaka Hosana (2019-04)

Thesis (MSc)--Stellenbosch University, 2019.

Thesis

ENGLISH ABSTRACT : This thesis investigates the relationship between the handle element of the De Rham cohomology algebra of a compact oriented smooth manifold, thought of as a Frobenius algebra, and the Euler class of the manifold. In this way it gives a complete answer to an exercise posed in the monograph of Kock [5] (which is based on a paper of Abrams [6]), where one is asked to show that these two classes are equal. Firstly, an overview of De Rham cohomology, Thom and Euler classes of smooth manifolds, Poincaré duality, Frobenius algebras, and their graphical calculus is given. Finally, it is shown that the handle element and the Euler class are indeed equal for even-dimensional manifolds. However, they are not equal for odd-dimensional manifolds.

AFRIKAANSE OPSOMMING : Hierdie proefskrif ondersoek die verband tussen die handvatselelement van die De Rham-kohomologie-algebra van ’n kompakte georiënteerde gladde variëteit, beskou as ’n Frobenius-algebra, en die Euler-klas van die variëteit. Op hierdie manier gee dit ’n volledige antwoord op ’n oefening wat in die monografie van Kock [5] gebaseer is (wat gebaseer is op ’n papier van Abrams [6]), waar een gevra word om te wys dat hierdie twee klasse gelyk is. Eerstens word ’n oorsig gegee van De Rham-kohomologie, Thom- en Euler-klasse van gladde manifolds, Frobenius algebras, en hul grafiese notasie word gegee. Ten slotte word getoon dat die handvatsel en die Euler-klas inderdaad gelyk is vie ewe-dimensionele variëteite. Maar hulle is nie gelyk nie vir onewe-dimensionele variëteite nie.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/105881
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