Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...a (co)homology theory? Or: Shut up and calculate.
Nonsense. Excision should be with the pair (X,A) not (X,U). I also forgot to add that \delta needs to be a natural transformation H_* to H_(*-1), i.e. a map between these functors. (This just means that certain squares commute.)
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.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algtop-2021.html
Eilenberg-Steenrod axioms.
https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms
https://www.jstor.org/stable/87896
Uniqueness Theorem.
https://msp.org/pjm/1962/12-1/pjm-v12-n1-p31-p.pdf
Various versions of “usual” homology.
https://en.wikipedia.org/wiki/Singular_homology
https://en.wikipedia.org/wiki/Cellular_homology
https://en.wikipedia.org/wiki/Simplicial_homology
Various versions of “extraordinary” homology (the duals of the listed cohomology theories).
https://en.wikipedia.org/wiki/List_of_cohomology_theories
The various theorems mentioned.
https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem
https://en.wikipedia.org/wiki/Hairy_ball_theorem
(Linked at https://en.wikipedia.org/wiki/Cowlick)
Pictures used.
https://en.wikipedia.org/wiki/Sphere#/media/File:Sphere_wireframe_10deg_6r.svg
https://mathimages.swarthmore.edu/index.php/File:Coffee112.jpg
https://demonstrations.wolfram.com/TheHairyBallTheorem/
Hatcher’s book (I sometimes steal some pictures from there).
https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Always useful.
https://en.wikipedia.org/wiki/Counterexamples_in_Topology
Mathematica.
https://mathworld.wolfram.com/Eilenberg-SteenrodAxioms.html
https://demonstrations.wolfram.com/TheHairyBallTheorem/
#algebraictopology
#topology
#mathematics