To be precise, let $M$ and $N$ be two 2n-dimensional smooth, closed manifolds that are homeomorphic. If $M$ admits an almost complex structure, can we deduce that $N$ also admits an almost complex structure?

  • 13
    $\begingroup$ In dimensions $\leq 8$, we have a complete set of obstructions to the existence of an almost complex structure, and one can check that they don't depend on the smooth structure. I believe the answer is not known in general, but I would be happy to be proved wrong. $\endgroup$ Jun 6 at 13:04
  • $\begingroup$ @MichaelAlbanese: do you happen to know if there homotopy equivalent closed smooth manifolds one of which is almost complex and the other one isn't? $\endgroup$ Jun 7 at 3:04
  • 5
    $\begingroup$ @IgorBelegradek, Kahn has shown that for every $k\geq1$ there exist a pair of smooth, closed, connected, oriented $8k$-manifolds, which are oriented homotopy equivalent, but only one of which admits an almost complex structure. See Corollary 6 of Obstructions to extending almost X-structures. $\endgroup$
    – Tyrone
    Jun 7 at 7:44
  • 2
    $\begingroup$ On the first page of Kahn's paper he states that $CP^2$ with the opposite orientation admits no almost complex structure. This answers the original question. Of course, one can then ask a modified question with "homeomorphism" replaced by "orientation-preserving homeomorphism". $\endgroup$ Jun 7 at 11:04


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