Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:


1. If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a [blog posting by David Speyer](http://sbseminar.wordpress.com/2008/02/14/complex-manifolds-which-are-not-algebraic/), you still have $H^2(M,\mathbb{R}) \ne 0$ even if $M$ is non-projective but algebraic.)
2. An interesting first example of a non-Kähler manifold is a Hopf manifold, by definition $(\mathbb{C}^n\setminus 0)/\Gamma\_r$, where $\Gamma\_r$ is a rescaling by $r$ with $|r| \ne 0,1$. This example has $H^1(M,\mathbb{R}) \ne 0$.
3. On the other hand, even-dimensional, compact Lie groups have left-invariant complex structures. If $M$ is such a manifold and is simply connected, then it is also 2-connected. $H^1(M,\mathbb{Z}) = H^2(M,\mathbb{Z}) = 0$ and $M$ is manifestly not Kähler. On the other hand, no such example is 3-connected and you always have $H^3(M,\mathbb{R}) \ne 0$.
4. There is (or was) a long-standing conjecture that no even-dimensional sphere other than $S^2$ has a complex structure.


So, question: Is there for each $n$, a compact, complex manifold $M$ which is $n$-connected?



Highly connected, compact complex manifolds
E. Calabi, B. Eckmann, *A class of compact, complex manifolds which are not algebraic.*
Ann. of Math. (2) 58, (1953). 494–500. 


From Chern's MR review (MR0057539): 



> 
> This paper defines on the topological product $S^{2p+1} \times S^{2q+1}$ of two spheres of dimensions $2p+1$ and $2q+1$ respectively, $p$ > 0, a complex analytic structure. The complex manifold so obtained ... admits a complex analytic fibering, with two-dimensional tori as fibers and having as base space the product $\mathbb{P}^p \times \mathbb{P}^q$ of complex projective spaces of (complex) dimensions $p$ and $q$ respectively.
> 


