In the mathematics literature, for example in Lee's Introduction to Smooth Manifolds, the singular homology groups of a smooth manifold $M$ are defined using singular $p$-chains, which are formal superpositions of singular $p$-simplices. For the purposes of this question I will define singular $p$-simplices as smooth maps from the standard $p$-simplices to $M$. By contrast, in the physics literature, for example "String Theory and M-theory", by Becker/Becker/Schwarz, chains are instead defined as formal superpositions of oriented compact submanifolds of $M$. Do these define the same singular homology (and cohomology) groups for all smooth manifolds $M$? I believe this reduces to the question of whether or not any oriented compact submanifold $N$ in $M$ can be tiled by the image of a set of singular $p$-simplices whose orientations are consistent with the orientation of $N$. Is this correct? If not, why not?



Singular Homology Groups from Compact Oriented Submanifolds?
Expanded version of my comment:


Please google "the steenrod realization problem". This is a foundational problem in algebraic topology. It has a nice answer -- Glen Bredon's book "Topology and Geometry" covers it pretty well.


Bredon's book is not the full story but it gets you quite close to it. 


For example if you use $\mathbb Z\_2$ coefficients then every homology class is realizable as a manifold. Similarly, if you use real coefficients (or rational) then a multiple of every homology class is realizable. 


The difficulty comes with odd torsion classes, when you use integral homology. Some of these classes are not realizable. Bredon's book gives you a good idea of how the machine works. 


Similarly, for integral homology it's known that in certain dimension ranges (both low and high dimensional) Steenrod realization holds. It's the middle-dimensional homology classes (or close to it) in odd torsion that are problematic.


Although I have not read Becker/Becker/Schwarz, my guess is they are using homology with real coefficients. This would explain why they are comfortable with singular manifolds and not the more conventional homology formalism with formal linear combinations of singular simplices, etc. 


edit: I suppose it should be noted that the more you look at this problem the more "layers" it has. If you demand your manifolds be genuine sub-manifolds and not singular maps from manifolds to a given manifold, then the problem is more subtle. Many of the results I mention above still hold. But generally speaking less is known at this level of specificity. If this is what you are interested in, the Bredon book gets you started on it as well. Serre gave a rather beautiful approach to this problem, which is what Bredon describes. 


