Preliminary exam prep
Here I'll present complexes from the most restrictive to the most general. Recall the standard n-simplex is
Δn={x∈Rn+1 : ∑xi=1,xi0}.
Definition: Let V be a finite set. A simplicial complex X on V is a set of distinct subsets of V such that if σ∈X, then all the subsets of σ are in X.
Every n-simplex in a simplicial complex is uniquely determined by its vertices, hence no pair of lower dimensional faces of a simplex may be identified with each other.
Definition: Let A,B be two indexing sets. A Δ-complex (or delta complex) X is
X=⨆α∈AΔnαα/{Fkββ}β∈B ,Fkββ={Δkβ1,…,Δkβmβ},
such that if σ appears in the disjoint union, all of its lower dimensional faces also appear. The identification of the k-simplices in Fk is done in the natural (linear) way, and restricting to identified faces gives the identification of the Fk−1 where the faces appear.
To define simplicial homology of a simplicial or Δ-complex X, fix an ordering of the set of 0-simplices (which gives an ordering of every σ∈X), define Ck to be the free abelian group generated by all σ∈X of dimension k (defined by k+1 0-simplices), and define a boundary map
∂k : Ck→Ck−1, [v0,…,vk]↦∑ki=0(−1)i[v0,…,^vi,…,vk].
Then Hk(X):=ker(∂k)/im(∂k+1).
Recall the standard n-cell is en={x∈Rn : |x|⩽, also known as the n-disk or n-ball.
Definition: Let X_0 be a finite set. A cell complex (or CW complex) is a collection X_0,X_1,\dots where
X_k := \left.X_{k-1}\bigsqcup_{\alpha\in A_k} e^k_\alpha \right/\left\{\partial e^k_\alpha\sim f_{k,\alpha}(\dy e^k_\alpha)\right\}_{\alpha\in A_k},
where the f_{k,\alpha} describe how to attach k-cells to the (k-1)-skeleton X_{k-1}, for k\>1. X_k may also be described by pushing out e^k\sqcup_{\dy e^k}X_{k-1}. Note that \dy e^k = S^{k-1}, the (k-1)-sphere.
To define cellular homology, we need more tools (relative homology and excision) that require a blog post of their own.
References: Hatcher (Algebraic topology, Chapter 2.1)
Δn={x∈Rn+1 : ∑xi=1,xi0}.
Definition: Let V be a finite set. A simplicial complex X on V is a set of distinct subsets of V such that if σ∈X, then all the subsets of σ are in X.
Every n-simplex in a simplicial complex is uniquely determined by its vertices, hence no pair of lower dimensional faces of a simplex may be identified with each other.
Definition: Let A,B be two indexing sets. A Δ-complex (or delta complex) X is
X=⨆α∈AΔnαα/{Fkββ}β∈B ,Fkββ={Δkβ1,…,Δkβmβ},
such that if σ appears in the disjoint union, all of its lower dimensional faces also appear. The identification of the k-simplices in Fk is done in the natural (linear) way, and restricting to identified faces gives the identification of the Fk−1 where the faces appear.
To define simplicial homology of a simplicial or Δ-complex X, fix an ordering of the set of 0-simplices (which gives an ordering of every σ∈X), define Ck to be the free abelian group generated by all σ∈X of dimension k (defined by k+1 0-simplices), and define a boundary map
∂k : Ck→Ck−1, [v0,…,vk]↦∑ki=0(−1)i[v0,…,^vi,…,vk].
Then Hk(X):=ker(∂k)/im(∂k+1).
Recall the standard n-cell is en={x∈Rn : |x|⩽, also known as the n-disk or n-ball.
Definition: Let X_0 be a finite set. A cell complex (or CW complex) is a collection X_0,X_1,\dots where
X_k := \left.X_{k-1}\bigsqcup_{\alpha\in A_k} e^k_\alpha \right/\left\{\partial e^k_\alpha\sim f_{k,\alpha}(\dy e^k_\alpha)\right\}_{\alpha\in A_k},
where the f_{k,\alpha} describe how to attach k-cells to the (k-1)-skeleton X_{k-1}, for k\>1. X_k may also be described by pushing out e^k\sqcup_{\dy e^k}X_{k-1}. Note that \dy e^k = S^{k-1}, the (k-1)-sphere.
To define cellular homology, we need more tools (relative homology and excision) that require a blog post of their own.
References: Hatcher (Algebraic topology, Chapter 2.1)
No comments:
Post a Comment