- DEF k-simplex: convex hull of k+1 points in affinely independent
- DEF n points are affinely independent: \( x_2-x_1, x_3-x_1, ..., x_n-x_1\) are linearly independent
- Affinely independent implies no 3 points are colinear
- Affinely independent around x1 implies affinely independent around x_i
- Kind of gives you a perspective on the edges which can form in a k-simplex
- DEF face: simplex from subset of these points is called the face of the simplex
- DEF simplicial complex: set K of simplexs such that
- face of any simplex in K is in K
- interesection of 2 simplexes is a face in both of them
DEF Abstract Simplicial Complex is a set K of a fixed set F
- \(K = \{ S | S \subset F \}\)
- For \( A \in K\), \( S \subset A \implies S \in K\)
DEF k-chain: vector space over field \(\iZ / 2\iZ\) with basis given by k-simplexs of simplicial complex
- DEF K boundary map: \(b_k: C_k \rightarrow C_{k-1}\) given by linear map on basis, if k-1 simplex is a face of the k-simplex
- DEF k-cycles: x, k-chain such that \(b_k(x)=0\)
DEF k-boundaries: \(x \in IM(b_{k+1})\)
\(IM(b_{k+1}) \subset KER(b_{k})\)
convex hull can be formed with any points
convex hull is unique up to permutation of the points
there are k unique k-1 face of a k simplex (each face excludes 1 point)
You can write the k-1 faces of ABCD as
ABC, BCD, CDA, DAB
The consequence k-2 faces are
AB BC BC CD CD DA DA AB
This shows \(IM(b_{k+1})\) is in kernel of \(b_{k}\)
DIM(A/B) = DIM(A)-DIM(B)
A L. independent basis that generates a vector space is independently maximal
- For any LI basis, the subset of basis is also LI. Look at a single additional element
- additional basis element can be generated by the old basis. then coefficient old basis representation and -1 of new basis element is 0.
DEF k-homology: \(Z_k / B_k \) where \(Z_k\) is \(KER(b_k)\) and \(B_k\) is \(IM(b_{k+1})\)
- DEF k-betti number: \(DIM( Z_k / B_k)\)