Can (singular) homology classes always be represented by images of closed manifolds?

algebraic-topology homology-cohomology manifolds

Solution 1:

This is a classical problem in algebraic topology, the Steenrod problem, which was more or less solved by Thom. Thom showed that this is true

  • $\bmod 2$,
  • rationally, and
  • integrally if $k \le 6$.

Integrally for $k \ge 7$ there are counterexamples; the obstructions involve Steenrod operations for odd primes. (It's harder to glue a bunch of simplices into a manifold than you're making it sound!) See this MO question and this MO question for more details, as well as this overview of Thom's work by Sullivan.

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