Topology

Let \(X\) be a topological space and \(x_0 \in X\text{,}\) and let \(n \geq 1\) be a natural number. The \(n\)-th homotopy group of \(X\) at \(x_0\text{,}\) denoted \(\pi n(X,x_0)\text{,}\) is the set of homotopy classes of maps \((In,Bd\R^n(In)) \to (X,x_0)\) with group operation defined by \([f][g] = [f ∗ g]\) for all \([f],[g] \in \pi n(X,x_0)\text{.}\)

In \(\R^n+1\text{,}\) the \(i\)-th standard basis vector is \(\varepsilon i(n) := (0,0,\dots,0,1,0,\dots,0)\text{,}\) where the coordinates are indexed by \(0,\dots,n\) and the \(1\) appears in the \(i\)-th coordinate.

The standard \(n\)-simplex is the subspace of \(\R^n+1\) (with the Euclidean topology) given by \(\delta n :=\{\sum i=0n ti\varepsilon i(n) | \sum i=0n ti = 1 \text{ and }0 \leq ti \leq 1 \text{ for all }i\}\text{.}\)

The boundary of the standard \(n\)-simplex is the subspace \(∂\delta n :=\{\sum i=0n ti\varepsilon i(n) | \sum i=0n ti = 1, 0 \leq ti \leq 1 \text{ for all }i, \text{ and }tj = 0 \text{ for some }j\}\text{.}\) The open \(n\)-simplex is the subspace \(0\delta n := \delta n - ∂ \delta n\) for \(n \geq 1\text{,}\) and \(0\delta 0 := \delta 0\text{.}\)

For \(0 \leq i \leq n\text{,}\) the \(i\)-th face map of the standard \(n\)-simplex is the function \(Fi(n): \delta n-1 \into \delta n\) defined by \(Fi(n)(\sum i=0n-1 ti\varepsilon i(n-1)) := \sum j=0i-1 tj\varepsilon j(n) + \sum j=i+1n tj-1\varepsilon j(n)\text{.}\)

A \(\delta\) -complex is a topological space \(X\) constructed inductively by: Let \(X(0)\) be a set of points with the discrete topology. Write \(X(0) = ∐\alpha \in J0 \delta \alpha 0\text{,}\) where for each \(\alpha, \delta \alpha 0 \cong \delta 0\text{.}\) For each \(\alpha \in J0\text{,}\) let \(\sigma ̄\alpha 0: \delta \alpha 0 \to X(0)\) be the inclusion map.

For each \(n \geq 1\text{,}\) form the space \(X(n)\) from \(X(n-1)\) by: For each \(\alpha\) in an index set \(Jn\text{,}\) let \(\delta \alpha n\) be a copy of the standard \(n\)-simplex \(\delta n\text{,}\) and let \(s\alpha : ∂\delta \alpha n \to X(n-1)\) be a continuous function such that for all \(0 \leq i \leq n\text{,}\) \(s\alpha \circ Fi(n)|∂\delta n = \sigma ̄\beta n-1\) for some \(\beta \in Jn-1\text{.}\) Let \(Zn := X(n-1) ∐ (∐\alpha \in Jn \delta \alpha n)\) have the disjoint union topology, and let \(\sim\) be the minimal equivalence relation on this space such that \(d \sim s\alpha (d)\) for all \(d \in ∂\delta \alpha n\) and \(\alpha \in Jn\text{.}\) Define \(X(n)\) be the quotient space and let qn be the quotient map. Define \(\sigma ̄\alpha n: \delta \alpha n \to X(n)\) to be the composition \(\sigma ̄\alpha n := qn \circ i\delta \alpha nZn\) where \(i\delta \alpha nZn: \delta \alpha n \to Zn\) is the inclusion map. Let \(X := \cup n \in \N X(n)\) (where each \(X(n)\) is identified with its image in \(X(n+1)\)). For all \(n\) and for all \(\alpha \in Jn\text{,}\) define \(\sigma \alpha n = \sigma \alpha : \delta \alpha n \to X\) by \(\sigma \alpha n := iX(n)X \circ \sigma ̄\alpha n\) where \(iX(n)X: X(n) \to X\) is the inclusion. A subset \(A\) of \(X\) is defined to be open in \(X\) if and only if \((\sigma \alpha n)-1(A)\) is open in \(\delta \alpha n\) for all \(n \geq 0\) and all \(\alpha \in Jn\text{.}\) Each \(\sigma \alpha n\) is called an \(n\)-simplex of \(X\text{.}\)

Let \(X\) be a \(\delta\) -complex and let \(n \geq 0\text{.}\) The group of simplicial \(n\)-chains of \(X\) is \(Cn\delta (X) := \oplus \alpha \in Jn \Z = \oplus n-simplices \sigma \alpha \Z \sigma \alpha\text{.}\) An element of \(Cn\delta (X)\) is called a (simplicial) \(n\)-chain, and written \(\sum \alpha \in Jn m\alpha \sigma \alpha\text{,}\) where each \(m\alpha \in \Z\) and all but finitely many \(m\alpha\)'s are \(0\text{.}\)

Let \(X\) be a \(\delta\) -complex and let \(n \geq 1\text{.}\) The \(n\)-th simplicial boundary map is the group homomorphism \(∂n\delta = ∂n: Cn\delta (X) \to Cn-1\delta (X)\) defined by \(∂n(\sum \alpha \in Jn m\alpha \sigma \alpha ) := \sum \alpha \in Jn m\alpha ∂n(1\sigma \alpha )\) and \(∂n(1\sigma \alpha ) := \sum i=0n (-1)i \sigma \alpha \circ Fi(n)\text{.}\) The \(0\)-th simplicial boundary map is the group homomorphism \(∂0\delta = ∂0: C0\delta (X) \to 0\text{.}\)

Let \(X\) be a \(\delta\) -complex and let \(n \geq 0\text{.}\) The \(n\)-th simplicial homology group of \(X\) is the group \(Hn\delta (X) := Ker ∂n\delta / Im ∂n+1\delta\text{.}\)