Locally finite homotopy types and rational cardinality

A homotopy type $X$ is locally finite if every homotopy group ${\pi }_i(X,x)$ is finite for every point $x$, and for each $x$ there is an integer $n_x$ such that ${\pi }_i(X,x)=0$ for $i>n_x$.

Write ${\pi }_0(X)$ for the set of connected components. Let ${\mathbb{Q}}_c(X)$ be the rational-valued functions on ${\pi }_0(X)$ with finite support. For a component $x$, the homotopy weight is

The product is finite because the positive homotopy groups vanish in high degrees; a negative exponent means a reciprocal.

For a map $f:X\to Y$ of locally finite homotopy types and $\alpha \in {\mathbb{Q}}_c(X)$, define

where $F_y$ is the homotopy fiber of $f$ at $y$ (Homotopy fiber of a map) and $i_y:F_y\to X$ is its natural map. For a proper map, meaning one whose homotopy fibers have finitely many connected components, also define $f^{\ast }:{\mathbb{Q}}_c(Y)\to {\mathbb{Q}}_c(X)$ by $f^{\ast }\beta =\beta \circ {\pi }_0(f)$.

The ratio lemma Homotopy-fiber weights reduce to componentwise ratios ↗ proves that fibers of maps between locally finite homotopy types are locally finite and that the displayed sum is finite. The set of components of a homotopy fiber alone does not determine $f_{!}$.

For an arbitrary map $f:X\to Y$ and a chosen point $y\in Y$, the unbased homotopy fiber used here is the mapping-path fiber $F_y=\{(x,\gamma )\in X\times Y^I:\gamma (0)=f(x),\gamma (1)=y\},$ with the path-space and subspace topology of Homotopy fiber of a map, forgetting its required basepoint. No point of $X$ mapping to $y$ is required. If $x_0\in X$ satisfies $f(x_0)=y$, the based construction in Homotopy fiber of a map is this same space with the basepoint $(x_0,c_y)$. If $\eta$ is a path from $y$ to $y^{\mathrm{\prime }}$, the map $(x,\gamma )\mapsto (x,\gamma \ast \eta )$ has homotopy inverse $(x,{\gamma }^{\mathrm{\prime }})\mapsto (x,{\gamma }^{\mathrm{\prime }}\ast {\eta }^{-1})$, using path reparameterization and cancellation of $\eta \ast {\eta }^{-1}$. Thus the fiber homotopy type depends only on the chosen target component.