What are...homotopy groups?

Goal.
Explaining basic concepts of algebraic topology in an intuitive way.

This time.
What are...homotopy groups? Or: Spheres in spaces.

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

Homotopy groups.
https://en.wikipedia.org/wiki/Homotopy_group
https://www.mathphysicsbook.com/mathematics/algebraic-topology/counting-the-ways-a-sphere-maps-to-a-space/the-higher-homotopy-groups/

Fundamental groups.
https://en.wikipedia.org/wiki/Fundamental_group
https://math.unice.fr/~indira/Mygifs.html

Eckmann-Hilton argument.
https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument
https://wildtopology.wordpress.com/2020/08/02/infinite-commutativity-part-i/
https://ncatlab.org/nlab/show/Eckmann-Hilton+argument
https://math.stackexchange.com/questions/1203807/applications-of-eckmann-hilton-argument/1203969
https://www.archimedestub.com/wp-content/uploads/2019/09/Eckmann-Hilton.pdf

Homotopy groups of spheres.
https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
https://blogs.scientificamerican.com/roots-of-unity/higher-homotopy-groups-are-spooky/

Pictures used.
https://math.unice.fr/~indira/GIFS/Pi1.gif
https://www.mathphysicsbook.com/wp-content/uploads/2013/01/31.higher-homotopy.png
https://www.archimedestub.com/tag/eckmann-hilton-argument/
https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres

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/HomotopyGroup.html

#algebraictopology
#topology
#mathematics

Видео What are...homotopy groups? канала VisualMath