Topology
Point Set Topology
Munkres
Homeomorphism
https://en.wikipedia.org/wiki/Homeomorphism https://en.wikipedia.org/wiki/Simplicial_complex_recognition_problem
https://en.wikipedia.org/wiki/Triangulation_(topology)
Brouwer fixed point https://en.wikipedia.org/wiki/Sperner%27s_lemma
jordan curve theorem
covering spaces
fundamental polyhedron  polyhedra schema. Those diagram with glued edges. The edges are generators
Algebraic Topology
Peter May  Algebraic Topology a concise course
Complexes
 simplicialcomplex  maybe the kind of obvious one. Triangulate your space.
 deltacomplex https://en.wikipedia.org/wiki/Delta_set delta set, semisimplicial set

CWcomplex  Bread and butter
 simplicial set https://en.wikipedia.org/wiki/Delta_set
 Kan complex
# toy around with hap definitions
class CW():
#triangle
points = [None,None,None]
edges = [(0,1), (1,2), (2,0)]
faces = [(0,1,2)] # indices refer to previous
# square
points = [None,None,None,None]
edges = [(0,1), (1,2), (2,3), (3,0)]
faces = [(0,1,2,3)] # indices refer to previous
Discrete finite represntation of CW uses loops over previous layers. The “disk” is a polygon. This is nice compared to simplicial because there is a nicer notion of “simplyfing” a space by removing redundant subdivision.
Regular vs non regular
Homotopy
Deformation Retract $f_t : X > X$ s.t. $f_0 = id$ and f_1 = A and f_t(A) = A for all t. It leaves A alone for all t. https://en.wikipedia.org/wiki/Retraction_(topology). Retraction is projection to subspace that preserves all points in subspace.
https://en.wikipedia.org/wiki/Inclusion_map
Homology
Morse Theory
Computational Topology
https://en.wikipedia.org/wiki/Computational_topology https://www.computop.org/ lots of cool programs
computatonal toploggy edelsbrunner harer notes
HAP GAP https://docs.gapsystem.org/pkg/hap/www/index.html graham ellis. Book https://www.youtube.com/watch?v=UMpTTuRdMA0&ab_channel=InstitutFourier
http://snappy.computop.org/ snappea for 3 manifolds. python bindings http://snappy.computop.org/verify_canon.html canonical retriangulation / cell decoomposition geometrization theorem dehnfilling There’s like a canonical geometrical surface? And then metrical things like volume are cannical?
uniformizaton
http://chomp.rutgers.edu/ chomp computational homology
exact reals? hott?
toploogical data anaysis persistent topology
seifert
computational group theory MAF https://maffsa.sourceforge.net/manpages/MAF.html KBMAG automata to describe the multiplication operation