group that $\ell$ represents.

As a consequence, we also have an algorithm that, given two equivariant simplicial maps $X\to Y$, tests whether they are equivariantly homotopic under the above dimension restrictions on $X$. Building on the methods of the present paper, [@Filakovsky:suspension] removes the dimension restriction for the latter question: it provides a homotopy-testing algorithm assuming only that $Y$ is simply connected.

The lifting-extension problem {#the-lifting-extension-problem .unnumbered}
-----------------------------

We obtain Theorem \[theorem:equivariant\] by an inductive approach that works more generally and more naturally in the setting of the *(equivariant) lifting-extension problem*, summarized in the following diagram: $$\label{eq:basic_square}
\xy *!C\xybox{\xymatrix@C=40pt{
A \ar[r]^-{f} \ar@{ >->}[d]_-\iota & Y \ar@{->>}[d]^-\varphi \\
X  \ar[r]_-{g} \ar@{-->}[ru]^-{\ell
} & B
}}\endxy$$ The input objects for this problem are the solid part of the diagram and we require that:

1.  $A$, $X$, $Y$, $B$ are free $G$-spaces;

2.  $f{\colon}A\to Y$ and $g{\colon}X\to B$ are equivariant maps;

3.  $\iota{\colon}A{\mathbin{\:\!\!\xymatrix@1@C=15pt{{}\ar@{ >->}[r]^{} & {}}}}X$ is an equivariant inclusion map;

4.  $\varphi{\colon}Y{\mathbin{\:\!\!\xymatrix@1@C=15pt{{}\ar@{->>}[r]^{} & {}}}}B$ is an equivariant *fibration* (which we mark by the double-head arrow)[^2]; and

5.  the square commutes (i.e. $g\iota=\varphi f$).

The lifting-extension problem asks whether there exists a *diagonal* in the square, i.e. an (equivariant) map $\ell{\colon}X\to Y$, marked 