Most models of singular chains on a topological space $X$ use maps from some particular collection of "nice" objects, such as the standard simplices $\Delta^n$, the standard cubes $[0,1]^n$, etc.
Is there a model of singular chains on a topological space $X$ which uses unparameterized maps from a similar such collection of nice objects? By unparameterized, I mean that a generator $\sigma:(\text{nice object})\to X$ is the same as the generator $\sigma\circ a$ for any $a\in\operatorname{Aut}(\text{nice object})$.
My motivation for this question is the following. If $\pi:E\to B$ is a principal $S^1$-bundle, I would like to have the Gysin map $\pi^!:H_\ast(B)\to H_{\ast+1}(E)$ defined on the chain level. Intuitively, I should send a simplex $\sigma:\Delta^n\to B$ to the associated map $\Delta^n\times S^1\to E$ (note that $\sigma^\ast E$ is trivial since $\Delta^n$ is contractible). Of course, there is no canonical trivialization, though, so this map $\Delta^n\times S^1\to E$ is only defined up to an automorphism of the domain. Hence the desire to have chains generated by unparameterized maps.