Steenrod asked the following question: can all singular homology classes be represented by manifolds, namely are they push-forward of fundamental classes of oriented smooth manifolds? Thom answered this question both positively and negatively: yes, for mod 2 and rational homology, but in generally no for integral homology. He observed there are infinitely many topological obstructions to “resolving the singularities” of a integral homology class, and constructed an example where those obstructions do not vanish.

However, Hironaka later showed, using machinery from algebraic geometry, that all complex algebraic varieties admit resolutions. The topological consequence of that is, all those obstructions discovered by Thom must vanish on algebraic homology classes of a complex algebraic variety, which is quite surprising.

In this talk, I will try to explain what are those topological obstructions, and (if time permitted) show why those obstructions vanish for low-dimensional complex algebraic varieties without referring to Hironaka’s theorem.