Induced Homomorphism in Algebraic Topology
Definition 1 2
Let $X,Y$ be a topological space and $\varphi : X \to Y$ be continuous. The homomorphism defined as $$ \varphi_{\ast} [f] := \left[ \varphi f \right] = \left[ \varphi \circ f \right] \qquad , \forall f : I \to X $$ and $$ \varphi_{\ast} : \pi_{1} \left( X, x \right) \to \pi_{1} \left( Y, \varphi (x) \right) $$ is called Induced Homomorphism.
Theorem
- [1]: If both $\phi : X \to Y$ and $\psi : Y \to Z$ are continuous, then $\left( \psi \phi \right)_{\ast} = \psi_{\ast} \phi_{\ast}$ holds.
- [2]: If $\varphi : X \to Y$ is a homeomorphism, then $\varphi_{\ast} : \pi_{1} \left( X, x \right) \to \pi_{1} \left( Y, \varphi (x) \right)$ is an isomorphism.
Description
It’s natural for topology to be interested in homeomorphisms between two topological spaces, and for algebra to be interested in isomorphisms between two algebraic structures. That algebraic topology is curious about both is entirely natural, and the induced homomorphism plays a very significant role in the study of related theories.
Regarding notation, it’s easier to start by getting used to the form, rather than delving deep into how function composition works and the specifics of continuity as written in the definition. $$ \begin{align*} \phi :& X \to Y \\ \phi_{\ast} :& \pi_{1} X \to \pi_{1} Y \end{align*} $$ Looking at the format, we can see that the functions are distinguished by whether $\ast$ is attached or not. In the formulas above, excluding any explanation like $f : I \to X$ is a path in $X$, or $\varphi : X \to Y$ maps the starting point $x \in X$ of $X$ to $\varphi(x) \in Y$, or $[f] \in \pi_{1} (X, x)$ is a homotopy class, it’s simply summarized for easier memorization. In short,
- Without $\ast$, $\phi$ is about topology (continuous functions)
- With $\ast$, $\phi_{\ast}$ is about algebra (homomorphism).
A good way to remember this is by noting how operations between homotopy classes are denoted as $$ [f] \ast [g] = \left[ f \cdot g \right] $$ In algebraic topology, at least within the context of discussing homotopies, attaching $\ast$ always means it’s algebra-related, which makes it easier to study. Meanwhile, $f \cdot g$ is the Path Production of the endpoints of $f$ and $g$, and in the definition of $$ \varphi_{\ast} [f] := \left[ \varphi f \right] $$ $\varphi f$ represents the composition of functions, $\varphi \circ f$. This becomes clear upon some reflection because the codomain of $f$ is $X$, and the domain of $\varphi$ is $X$, making the composition natural and not merely a path, so the operation between them couldn’t possibly be a Path Product $\cdot$.