the case of all *continuous* maps $X\to Y$ up to homotopy, with no equivariance condition imposed, as in [@CKMSVW11], the abelian group structure on $[X,Y]$ is canonical. In contrast, in the setting of the lifting-extension problem, the structure is canonical only up to a choice of a zero element; in other words, $[X,Y]_B^A$ really has an “affine” nature (in very much the same way as an affine space is naturally a vector space up to a choice of its origin).

This non-canonicality of zero is one of the phenomena making the equivariant problem (and the lifting-extension problem) substantially different from the non-equivariant case treated in [@CKMSVW11]. We will have to devote much effort to dealing with the choice of zero, and working with “zero sections” in the considered fibrations.

Embeddability and equivariant maps {#embeddability-and-equivariant-maps .unnumbered}
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Theorem \[theorem:equivariant\] has the following consequence for embeddability of simplicial complexes:

\[t:emb-metast\] Let $n$ be a fixed integer. Let $n$ be an integer. There is an algorithm that, given a finite simplicial complex $K$ of dimension $k \leq \frac23 n-1$, decides the existence of an embedding of $K$ into ${\mathbb R}^n$ in polynomial time.

The algorithmic problem of testing embeddability of a given $k$-dimensional 