📂Topological Data Analysis

Retracts in Topology

Definition ^{1 2}

Let’s consider a subspace $A \subset X$ of a topological space $X$ and denote the identity function by $\text{id}$.

1. If there exists a continuous surjective function $r : X \to A$ that satisfies $$r \circ i = \text{id}_{A} : A \to A$$ for an inclusion $i : A \to X$, then $r$ is called a Retraction, and $A$ is called a Retract of $X$. In other words, $r$ is a continuous surjective function that satisfies the following. $$r(a) = a \qquad , \forall a \in A$$
2. If there exists a retraction $r : X \to A$ that satisfies the following, then $A$ is called a Deformation Retract of $X$. $$i \circ r \simeq \text{id}_{X} : X \to X$$ This means that $A$ being a deformation retract of $X$ means that there exists a homotopy $F : X \times I \to X$ as follows. $$\begin{align*} F(x,0) =& x & , x \in X \\ F(x,1) \in & A & , x \in X \\ F(a,t) =& a & , a \in A \end{align*}$$

Explanation

A Retract is a mapping that, true to its meaning, somehow crumples the entire space $X$ into a smaller space $A$ while keeping the original points $a \in A$ intact. What needs attention in the definition of a Deformation Retract is that it is not $i \circ r = \text{id}_{X}$ but $i \circ r \simeq \text{id}_{X}$, meaning it is homotopically equivalent.

By thinking about such retracts, we can now look at infinitely complex and diverse spaces in a very simple way. For example, whether it is a circle $X$ with thickness or a circle $A$ without, if there’s no difference in the properties of the space itself under consideration, it would be much easier to handle the latter.

Strong Deformation Retract

A deformation retract is considered when there exists a $r$ that satisfies $i \circ r \simeq \text{id}_{X}$ regardless of $A$, but if there exists a retraction $r$ that satisfies $$i \circ r \simeq_{\text{rel} A} \text{id}_{X}$$ considering relative homotopy as well, then $A \subset X$ is called a Strong Deformation Retract. This requirement might seem too stringent because it requires points in $A$ to be fixed, and indeed, it is not considered as important compared to a deformation retract.