Can we use unparameterized chains to calculate singular homology? 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.
 A: I write this as an answer because it is a bit longer than a comment.
1.If $X$ is a subset of some $\newcommand{\bR}{\mathbb{R}}$ Euclidean space $\bR^n$ which additionally is a locally Lipschitz  retract of one of its neighborhood, then  you can use  integral  flat currents to represent cycles  in this fashion.   
This is not quite what you want because the objects representing such  such chains are  rectifiable  sets  which can be quite wild.  Additionally,  the maps bewteen your topological spaces need to be locally Lipschitz. 
2.If the category of spaces you are interested in is smaller,  subanalytic sets  or, more generally, sets definable in some $o$-minimal  category, then   you can work with a smaller collection of chains. (All such sets    live in some finite dimensional Euclidean space.) As generators of your chain complex you can use the currents of integration  defined by the  oriented   real analytic  manifolds $S\subset X$.    Such submanifolds can be triangulated  and everything    is generated by the currents of integration along  simplices $\Delta\subset X$ such that their interiors are  smooth  manifolds. The Gysin map is simple in this case: it sets the current defined by $\Delta$ to the current defined by $\pi^{-1}(\Delta)$. This may seem like a restrictive category, but many spaces that admit  local Kuranishi-type descriptions      belong to this category, especially if they are compact.  Hardt  has the  complete description of this approach, though it may be  difficult to digest at first.
3.Sometime the  high-brow road may pay dividends. By that I mean thinking of homology  as cohomology with  coefficients in the dualizing  complex. Many   operations is can be emulated in this language.   Have a look at Iversen's "Cohomology of sheaves". It might give you some ideas. 
