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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.

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  • $\begingroup$ If you are willing to switch to cohomology, there is integration along fibers en.wikipedia.org/wiki/Integration_along_fibers. Perhaps something similar exist in singular homology. $\endgroup$
    – Mark Grant
    Feb 21, 2014 at 7:15
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    $\begingroup$ You could trivialize your question by artificially making all your "nice objects" have trivial automorphism group. $\endgroup$ Feb 21, 2014 at 9:42
  • $\begingroup$ @RyanBudney: but then this seems unsuitable for the intended application. $\endgroup$ Feb 21, 2014 at 17:18
  • $\begingroup$ @MarkGrant: Is there a way of defining integration along fibers for a model of cochains which is functorial under continuous maps? (if so, that would probably solve my problem) $\endgroup$ Feb 21, 2014 at 17:20
  • $\begingroup$ I'm not sure. There are various ways of defining forms on spaces which aren't manifolds; some of these are surveyed in section 7 of Sullivan's paper "Infinitesimal computations in topology". These should be functorial for continuous maps. $\endgroup$
    – Mark Grant
    Feb 22, 2014 at 17:08

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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.

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