It seems that whether a simply connected 4 manifold needs 1-handles and 3-handles is still an open question, see [Existence of Morse functions on simply connected manifolds](https://mathoverflow.net/questions/143790/existence-of-morse-functions-on-simply-connected-manifolds).


I am wondering if it is true that every Fermat surface $x^n+y^n+z^n+w^n=0$ admits a handle decomposition with only, 0,2,4 handles.



Existence of perfect Morse functions on Fermat surfaces $x^n+y^n+z^n+w^n=0$
Yes, this is true for any nonsingular hypersurface in ${\mathbb C}P^3$. See Harer, On handlebody structures for hypersurfaces in C3 and CP3. Math. Ann. 238 (1978), no. 1, 51–58.


