* COMPARING OBJECTS

homeomorphism
ambient isotopy: continuous F:R^dx[0,1]->R^d, F(*,0)=identity on R^d, F(X,1)=Y, F(*,t) homeomorphism onto R^d
ambient isotopy => homeo, but not the reverse (ex: trefoil knot)
(isotopie: peut contracter un noeud (mais pas un tube))
hard to determine if 2 spaces are homeo
weaker: homotopy equivalence
homotopic maps f0 and f1:X->Y if exists continuous H:[0,1]xX->Y, H(0,*)=f0, H(1,*)=f1
X and Y have same homotopy type (homotopy equivalent) if exists continuous f:X->Y and g:Y->X such that compositions are homotopic to identities
ex: ball~point, circle~annulus
contractible (aka trivial): homotopy equivalent to a point
hard to prove homotopy equivalence. Special case:
Y c X, exists continuous H:[0,1]xX->X with H(0,x)=x, H(1,X)cY, H([0,1],Y)cY => homotopy equivalent
deformation retract: same with H(t,y)=y
comparing objects with a metric
isometry (bijection that preserves distances) (implies homéo)
distance:
embedded: Hausdorff distance (direct ou sup de la difference entre les fcts distance)
intrinsic: Gromov-Hausdorff. inf epsilon / exists epsilon-correspondance. inf Hausdorff for isometric embeddings (GH trop dur à calculer, plus tard on verra bornes inf PH)

* SIMPLICIAL COMPLEX

geometric simplex: convex hull of affinely independent pts
face: CH of subset (excluding empty simplex?) ex: 7 faces for a triangle
complex=finite collection of simplices, s=>face(s), intersection empty or common face
dimension (max), k-skeleton
underlying space |K|=union, induced topology from R^d
abstract simplicial complex with vertex set V: a set of subsets of V, includes singletons, stable by subset
subset=simplex, k+1 elements=dimension k (k-simplex)
geometric->abstract: vertex scheme (obvious)
abstract->geometric: geometric realization. Always exists in R^n (number of vertices). Actually R^(2d+1)
topology of |K| does not depend on the realization (homeomorphism)

* NERVE

open cover of topological space X by subsets Ui (resp closed cover)
nerve=abstract complex C(U), vertex set={Ui}, simplex if non-empty intersection
thm (good cover): finite open cover, all intersections are empty or contractible => X and C(U) homotopy equivalent
thm2: finite closed convex cover => homotopy equivalent
fundamental: encode homotopy type of continuous space as combinatorial object
ex: union of balls
filtration of finite simplicial complex: nested sequence of sub-complexes, empty ... K, add one simplex at a time. ~order on simplices that extends "inclusion".
can index by real values instead (constant on intervals)
function on vertices -> can define lower-star filtration (f(sigma)=max f(v) pour v dans sigma) filtration sublevel sets f^-1(]-inf,t])
ex: Cech (simplex appears at radius of minimal enclosing ball), Rips
rem: they are big, 2^n
relation Cech c Rips c Cech2 (better than 2 in Euclidean)

* MAPPER
start with clustering = dim 0 approximation. dim 1: mapper ?

* TRIANGULATION COMBINATOIRE

star(p)=simplices that contain p (not a complex -> closed star)
link(p)=faces of s in star(p), that do not contain p (is a complex)
pure k-complex: every simplex is a face of a k-simplex
boundary dK of pure k-complex: (k-1)-simplices that have exactly 1 coface (of dim k), and their faces -> subcomplex
combinatorial k-manifold: pure k-complex, link of interior vertices triangulated (k-1)-sphere, link of boundary vertices triangulated (k-1)-ball
triangulation of a finite point set P: geometric simplicial complex K with vertex set P and underlying space CH(P)
triangulation of a topological space X: simplicial complex K and homeomorphism h:|K|->X
lots of hard problems (undecidable if K homeomorphic to 5-sphere)