Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.

And in the Harper and Greenberg's book there exists a following statement :

The representation of $H_2(M^4)$ be embedded surfaces is an important open problem, wheren $M^4$ is a closed $4$-dimensional manifold.

(1) Here I want to know anything related with them.

(2) If $S$ is an embedded surface in $M^4$, then is it important to know the exact value $[S]$ in $H_2(M^4)$ ?

(3) If finding a surface representing $n$ in $H_2(M^4)$ is a problem, then what is the known result ?

(4) In the paper four-manifolds which admit ${\bf Z}_p \times {\bf Z}_p$ actions - McCooey (See Section 2 in page 2), he state Edmonds's result :

If ${\bf Z}_p \times {\bf Z}_p$ acts locally linearly, homologically trivially on a closed $M^4$ with $H_1(M)=0$ and $ b_2(M)\geq 1$, then each nonzero element has a fixed point set consisting of isolated points and $2$-spheres, and each sphere represents a nontrivial homology class.

Here the fixed surface $S^2$ represents $1$ or $-1$ ?

  • 2
    $\begingroup$ I can't make sense of your questions as currently stated. Maybe the minimal genus problem for surfaces in 4-manifolds is relevant? $\endgroup$ – Tim Perutz Nov 16 '12 at 16:14

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