Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...homology intuitively? Or: What is a hole?
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-algtop-2021.html
Homology.
en.wikipedia.org/wiki/Homology_(mathematics)
math.stackexchange.com/questions/40149/intuition-of-the-meaning-of-homology-groups
Holes.
blogs.scientificamerican.com/roots-of-unity/what-we-talk-about-when-we-talk-about-holes/
Torus and solid torus.
en.wikipedia.org/wiki/Torus
en.wikipedia.org/wiki/Solid_torus
Pictures used.
www.freepik.com/premium-photo/swimming-pool-inner-tube_6362072.htm
en.wikipedia.org/wiki/Doughnut#/media/File:Glazed-Donut.jpg
www.mathphysicsbook.com/wp-content/uploads/2013/01/24.triangulation.png
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/AT/AT.pdf
Always useful.
en.wikipedia.org/wiki/Counterexamples_in_Topology
Mathematica.
mathworld.wolfram.com/Homology.html
#algebraictopology
#topology
#mathematics
24 авг 2021