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Lecture 5 - Here Comes the Homology

Goals

Goals for today:

• Ch 2.4: Cycles and Boundaries

Linear Algebra Review & Homology

Definition: A field \((k,+,\cdot)\) is a set \(k\) with two operations \(+\) and \(\cdot\) such that for any \(a,b,c \in k\):

• Closure:

• Commutativity:

• Associativity:

• Identity:

• Inverses:

• Distributivity:

Examples:

Vector space

Definition: A vector space over a field \(k\) is a set \(V\) with vector addition and scalar multiplication such that

• Associative \(+\):

• Commutative \(+\):

• Identity

• Inverses:

• Scalar vs. field mult:

• Distributivity:

Examples:

Basis

Definition: A basis for a vector space \(V\) is a collection of vectors \(\{b_\alpha\}_{\alpha \in A}\) such that

• they are linearly independent and,

• they span \(V\).

Goal

Build a vector space from a simplicial complex!

\(p\)-Chains

Let \(K\) be a simplicial complex and fix a dimension \(p\). A \(p\)-chain is a finite formal sum of \(p\)-simplices in \(K\), written $$\alpha = \sum a_i \sigma_i

\[ \begin{align}\begin{aligned} ## Addition of chains\\$p$-chains are added component-wise: if $\alpha = \sum a_i \sigma_i$ and $\beta = \sum b_i \sigma_i$, then $\alpha + \beta =$ ## Chain group\\The collection of $p$-chains with addition is called the $p^\text{th}$-chain group (vector space), $C_p(K)$.\\**Some checks**\\- Associative $+$:\\- Commutative $+$:\\- Zero element\\- Inverses: ## Linear Transformations\\A linear transformation between two vector spaces $V$ and $W$ is a map $T:V\to W$ such that the following hold:\\- - \\Matrix representation: ## Boundary maps[^1]\\$$\begin{matrix} \partial _p: & C_p(K) & \to & C_{p-1}(K)\ & \sigma = [v_0,\cdots,v_p] & \mapsto & \sum_{j=0}^p[v_0,\cdots,\widehat{v_j}, \cdots v_p] \end{matrix}\end{aligned}\end{align} \]

Checking linearity

\[\partial_p(\alpha + \beta) = \partial_p(\alpha) + \partial_p(\beta); \hspace{.5in} \alpha = \sum a_i \sigma_i,\qquad \beta = \sum b_i, \sigma_i\]

Homework

• \(\partial_1([a,e]) =\)

• \(\partial_1([a,e] + [b,e]) =\)

• \(\partial_1([a,e] + [c,e] + [c,d] + [a,d]) =\)

• \(\partial_2([a,c,e] + [a,c,d]) =\)