Thom's theorem states that for every homology class $\alpha \in H\_{\*}(X)$ there exists an integer $k = k(\alpha)$ such that the class $k\, \alpha$ comes from the fundamental class of an orientable closed smooth manifold, where $X$ is an arbitrary topological space. To be on the safe side we assume that $X$ is a countable CW-complex.


Given a topological pair $(X,A)$, $A$ is closed, and a homology class $\alpha \in H\_{\*}(X,A)$. Is it possible to realize $k\, \alpha$ by a map from an orientable manifold with boundary, for $k$ sufficiently large?



Relative Steenrod's problem
Yes. It's quite general. If $h$ is any generalized homology theory (for example, oriented bordism) such that $h\_0(point)=\mathbb Z$ and such that $h\_n(point)=0$ for all $n<0$ then there is a map from $h$ to ordinary homology inducing an isomorphism $h\_0\to H\_0$; and after tensoring with $\mathbb Q$ this map $h\to H$ becomes surjective for all spaces and pairs. You can see this using the Atiyah-Hirzebruch spectral sequence with $E^2\_{p,q}=H\_p(X,A;h\_q(point)\otimes \mathbb Q)$, abutting to $h\_{p+q}(X,A)\otimes \mathbb Q$. All of the differentials are zero because they are essentially stable operations on rational homology, so the edge $E^2\_{p,0}$ survives to $E^\infty$


