AlgTop24: The fundamental group

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This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.

The fundamental groups of the disk and circle are described.

This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.

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How are the special properties of the polygon's corners represented by the path philosophy? Are there some issues with infinity or (in other words) discontinuity in use?

flexibartr

On minute 33, what happens if the polygon m is not convex?

alfredorestrepo

@tothemesosphere Equivalence classes rely on modern set theory, involving so-called ìnfinite sets', which does not hold together logically under close inspection. These issues are and will be addressed further in my MathFoundations YouTube series at my Insights in Mathematics channel, user: njwildberger.

njwildberger

Can anyone give the pdf sheet file of this video

phoeniximostion

Dear Professor, is there any definition for the addition of two paths?

newchland

@alfredorestrepo Good question, in that case the argument is more complicated, but can still be made. However I admit that it is considerably less simple.

unswelearning

@tothemesosphere It's an attempt to avoid complications of set theories.

Either there are difficulties with the notion of infinite number of objects collected under single name and then it's problematic to claim that the product is well defined (it doesn't matter how it's called: class, type, collection) or there are no difficulties then there is no need to depart from the standard terminology. At the level of rigor of this course it seems extensive to address these issues.

merement

first deform the not convex polygon in a convex one, and consider that any path (and path deformation) in a polygon is identifiable to a path (and path deformation) in the other polygon.

sergiorgio

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