- Point Set Topology
- Algebraic Topology
- Computational Topology
Point Set Topology
Brouwer fixed point https://en.wikipedia.org/wiki/Sperner%27s_lemma
jordan curve theorem
fundamental polyhedron - polyhedra schema. Those diagram with glued edges. The edges are generators
- simplicial-complex - maybe the kind of obvious one. Triangulate your space.
- delta-complex https://en.wikipedia.org/wiki/Delta_set delta set, semisimplicial set
CW-complex - 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
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/Computational_topology https://www.computop.org/ lots of cool programs
http://snappy.computop.org/ snappea for 3 manifolds. python bindings http://snappy.computop.org/verify_canon.html canonical retriangulation / cell decoomposition geometrization theorem dehn-filling There’s like a canonical geometrical surface? And then metrical things like volume are cannical?
http://chomp.rutgers.edu/ chomp computational homology
exact reals? hott?
toploogical data anaysis persistent topology
computational group theory MAF https://maffsa.sourceforge.net/manpages/MAF.html KBMAG automata to describe the multiplication operation