Geometric sets determined by chains (for integration and Stokes' theorem)

Question

I have asked a similar question on mathSE more than a year ago, which received no answers, only a few comments which did not really help me. I am now re-asking this question here but reformulated slightly since now I actually understand better what is it that I don't understand and am curious about.

Although answers on the SE network are not strictly for the asker but for a wider audience, for potential answerers who wish to help me in particular, I make notice of the fact that I am well-educated in differential geometry, but less so in (algebraic) topology and measure theory, which means that

1. I am looking for an answer that does not involve heavy measure theory;
2. the question might be trivial or obvious to someone with a proper background in algebraic topology, but I don't have that background and reading alg top literature does not seem particularly helpful to me. I make this notice because I suspect that what I am looking for is basically the $C^\infty$ and slightly more degenerate version of a simplicial complex, but looking at simplicial complexes in alg top books don't really help me.

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The short summary of my problem is that differential geometry texts usually consider integration of differential forms and Stokes' theorem on (smooth) manifolds, manifolds with boundary and (sometimes; see eg. Lee's book) manifolds with corners.

However often actually quite simple domains of integration such as solid cones in $\mathbb R^3$ do not fit even into the class of manifolds with corners. And the theory of manifolds with corners seem awfully complicated! Stokes' theorem is nonetheless valid on cones.

So in short I am interested in defining a class of subsets of (smooth) manifolds on which differential forms can be integrated and Stokes' theorem holds and such that it is more general than manifolds with corners.

I know that it is possible to do this by using heavier machinery from measure theory and analysis than it is usual in differential geometry (the books by the geometric measure theory crowd for examples that I don't understand and Sauvigny - Partial Differential Equations for a less general example that I do understand but find inconvenient).

Instead, I want to follow an approach that uses chains. Chains are computation-friendly (they are already parametrized) and flexible (the aforementioned cone can be described for the purposes of integration as a single chain element), but has the drawback that - in a heuristic meaning - they are not "geometric". Integration over chains is integration over maps and it is - at least to me - not a trivial question when does a chain describe a subset of a manifold in a way that the underlying parametrizations are irrelevant (i.e. the integral of forms on the chain depends only on the subset determined by the chain and not on the particular representation in terms of chains).

Just to make it clear, I am not looking for results regarding the triangulatability of manifolds/manifolds-with-boundary/manifolds-with-corners, but rather I want to define a class of subsets of smooth manifolds that can be represented by chains for the purposes of integration via putting appropriate constraints on the representing chains.

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In greater detail, a $k$ dimensional (singular, smooth) chain element in a (smooth, real) $m$ dimensional manifold $M$ is a smooth mapping $\sigma:I^k\rightarrow M$, where $I^k\subseteq\mathbb R^k$ is the unit cube in $\mathbb R^k$, and a (singular, smooth) $k$-chain in $M$ is an element of the free $\mathbb Z$-module generated by (singular, smooth) $k$ dimensional chain elements, that is a formal linear combination $$ \sigma=\sum_{i=1}^N n_i\sigma_i $$of chain elements with integer coefficients. This $\mathbb Z$-module is denoted $\mathscr C_k(M)$, and it might be useful to introduce the quotient module $\tilde{\mathscr{C}}_k(M)=\mathscr C_k(M)/\sim$ where $\sim$ is the equivalence relation whereby two chains are equivalent if and only if all differential forms integrate the same way on them (call this the equivalence of chains).

The boundary of a $k$-dimensional chain element $\sigma:I^k\rightarrow M$ is the $k$-chain $$ \partial\sigma:=\sum_{i=1}^k\sum_{\alpha=0,1}(-1)^{i+\alpha}\partial^{(i,\alpha)}\sigma, $$where the $(i,\alpha)$-face $\partial^{(i,\alpha)}\sigma$ of $\sigma$ is defined by $$ \partial^{(i,\alpha)}(x^1,\dots,x^{i-1},x^{i+1},\dots,x^k)=\sigma(x^1,\dots,x^{i-1},\alpha,x^{i+1},\dots,x^k). $$

The boundary $\partial\sigma$ of a $k$-chain $\sigma=\sum_{i=1}^{N}n_i\sigma_i$ is given by extending $\partial$ to be a module homomorphism.

The set determined by a $k$-chain $\sigma$ is the subset $S(\sigma)\subseteq M$ given by $$ S(\sigma)=\bigcup_{i=1}^N\sigma_i(I^k),\quad\sigma=\sum_{i=1}^Nn_i\sigma_i. $$

I am looking for a reasonable set of conditions to be imposed upon $k$-chains such that the following holds (in the following all chains satisfy these conditions):

1. The set $S(\sigma)$ determined by $\sigma$ can be consistently given an orientation, such that $S(\sigma)$ and $-S(\sigma):=S(-\sigma)$ are the only two possibilities. In other words, no matter how $S(\sigma)$ is represented by a different chain $\bar\sigma$, its orientation is either the same what it inherits from $\sigma$ or the opposite.
2. If $\omega\in\Omega^k(M)$ is a smooth $k$-form on $M$ (but should also be valid for continuous $k$-forms too at least), $\sigma,\bar\sigma\in\mathscr C_k(M)$ are $k$-chains (that satisfy the yet-to-be-determined conditions) such that $S(\sigma)=S(\bar\sigma)$ (the sets determined by the chains agree), then $$ \int_\sigma\omega=\pm\int_{\bar\sigma}\omega. $$ The positive sign is valid if $\sigma$ and $\bar\sigma$ determine the same orientation for $S(\sigma)$ and the negative sign is valid if they determine the opposite. In other words - up to orientation - differential forms can be integrated on $S(\sigma)$ and the value of such integrals are independent of the way $S(\sigma)$ is represented in terms of chains (as long as the orientations are respected).
3. If $\sigma\in\mathscr C_k(M)$ is a $k$-chain that satisfies the pertinent conditions, then so does its boundary $\partial\sigma$, possibly up to equivalence of chains (for example we might need to throw away chain elements that are so singular their images degenerate to subsets whose $k-1$-dimensional contents are zero thus all $k-1$-forms integrate to $0$ on them). The point of this point is that the set $S(\partial\sigma)$ determined by the boundary should intuitively correspond to the boundary of $S(\sigma)$ and thus Stokes' theorem becomes valid - through the Stokes theorem for chains - for sets of the form $S(\sigma)$.

I cannot state with any certainty which are the pertinent conditions to be imposed on chains such that the above points hold, but I think at least some subset of the conditions should be

1. A $k$-chain $\sigma=\sum_{i=1}^N n_i\sigma_i$ should not have any nontrivial multiplicities, i.e. $n_i=1$ or maybe $n_i=\pm 1$.
2. Each chain element $\sigma_i$ should be an embedding (or maybe an injective immersion) when restricted to the interior $\mathrm{int}(I^k)$.
3. Each pair of chain element $\sigma_i,\sigma_j$ should be nonintersecting on the interiors of the unit cubes, i.e. $\sigma_i(\mathrm{int}(I^k))\cap\sigma_j(\mathrm{int}(I^k))=\varnothing$. They can be intersecting on the boundaries.
4. If $\sigma_i$ and $\sigma_j$ are a pair of chain elements that intersect, i.e. $\sigma_i(I^k)\cap\sigma_j(I^k)\neq\varnothing$, then their intersections should be faces with opposite orientations. In other words there exists an $(l,\alpha)$ and a $(r,\beta)$ such that $\partial^{(l,\alpha)}\sigma_i(I^{k-1})=\partial^{(r,\beta)}\sigma_j(I^{k-1})$ and $(-1)^{l+\alpha}=-(-1)^{r+\beta}$ and the two chain elements do not intersect anywhere else.

The purpose of these conditions is as follow.

• Condition 1 is necessary (but probably not sufficient) to ensure that $S(\sigma)$ does not depend on the representing chains, since if we increase the multiplicity of chain elements, the integrals of forms get multiplied but the underlying sets remain the same. Thus multiplicities have to go.
• Condition 2 ensures that the sets determined by each chain element in the sum is not "too degenerate" while still allowing flexibility (stuff like a ball, a sphere or a solid cone can be represented each by a single chain element that is an injective immersion (possibly an embedding I am not totally sure right now) on the interiors but is singular on the boundaries).
• Condition 3 further ensures that multiplicities are absent by disallowing nontrivial intersections of chain elements.
• Condition 4 is to ensure that only the outer boundaries of chain elements contribute to the total boundary. I think but I am also unsure that this condition is also sufficient to guarantee that each $S(\sigma)$ has a well-defined orientation, since this rigid matching of orientations on the boundaries should mean that the subset determined by the chain has only two possibilities instead of allowing "piece-by-piece" choices of orientations.

Questions:

I wish to know whether the conditions 1-4. I have outlined above are sufficient for the purposes I have outlined in the text. If not, what further conditions do I need? Are there superflouous conditions that are unnecessary or overly restrictive?

I would also appreciate references to textbooks or papers where this sort of integration theory on manifolds is detailed. A big reason why I am asking this question in the first place is that the proofs (for example that the orientation of $S(\sigma)$ is well-defined and there are only two of them, or that the integration only depends on $S(\sigma)$ and its orientation) are beyond me. Rather than just knowing whether what I did is correct or not, I'd also like to look at proofs.