2.4 G-spaces and coverings. Geometria III.

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Published at : 15 Dec 2021
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Given a topological space X, a "covering" is a topological space Y equipped with a surjective map onto X that "homotopically simplifies" X, in a sense. The simplification only regards the homotopical structure of X in dimension 1, leaving unchanged the higher ones. Coverings are closely related to certain group actions on topological spaces. In Chapter 3 of this course we will classify all the coverings of decent topological spaces.

Dato uno spazio topologico X, un suo rivestimento è uno spazio che Y con una mappa suriettiva su X che in un certo senso "semplifica omotopicamente" X. La semplificazione riguarda solo la struttura omotopica di X in dimensione 1, mentre lascia invariata quella in dimensione superiore.
La mappa determina un ricoprimento aperto di Y, in cui ogni aperto è l’unione disgiunta di componenti ciascuna omeomerfa alla sua immagine in X.
I rivestimenti hanno legami profondi con azioni di gruppi su spazi topologici e la loro classificazione sarà l’argomento delle lezioni del capitolo 3 di questo corso.

** rispetto al video 2.4 precedente: correzione nella definizione di rivestimento

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