TITLE: Cohomology and fundamental classes
QUESTION [70 upvotes]: Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup generated?

REPLY [2 votes]: According to an article in The Geometry and Topology of Submanifolds and Currents, edited by Weiping Li and Shinshu Walter Wei
The lack of suitable representatives in smooth homology by submanifolds was 
'partial motivation' for Federer to introduce rectifiable homology by defining 
• the chain groups as:
$Z_{i}(A,B):=\{T\in R_{m}(\mathbb{R}^{n},\mathbb{Z}): \partial T\in R_{m-1}(\mathbb{R}^{n},\mathbb{Z}),spt(T)\subset A$ and $spt(\partial T)\subset B\}$
• the boundary groups as 
$B_{i}(A,B):=\{\partial T:T\in Z_{i+1}(A,A)\}$
And the homology groups as usual by $H_{i}(A,B):=Z_{i}(A,B)/B_{i}(A,B)$
(Here, $A$ & $B$ are both compact Lipschitz neghbourhood retracts (of $\mathbb{R}^{n}$) and we have $B\subset A$). 
He showed that it satisfied the Eilenberg-Steenrod axioms, and hence is isomorphic to singular homology; and that every homology class in this homology contains a mass-minimising rectifiable representative. Thus we should see Federers notion of rectifiable currents as an appropriate generalisation of submanifold which allows us to identify homology classes with special sub-manifolds.