$c_1^3$ of 6 manifolds

Question

For a closed oriented smooth 4 manifold $X$ we have $c_1^2(X) := 2e(X) + 3σ(X)$, $c_1$ is the first Chern class and $σ$ is the signature. For 6 manifolds is there such a relation with $c_1^3$? I'd greatly appreciate any reference.

Answer 1

See the accepted answer here: https://mathoverflow.net/questions/63439/how-can-we-detect-the-existence-of-almost-complex-structures

In particular this paragraph:

A closed oriented 6-dimensional manifold $X$ without 2-torsion in $H^3(X,\Bbb Z)$ admits an almost complex structure. There is a 1-1 correspondence between almost complex structures on X and the integral lifts $W\in H_2(X,\Bbb Z)$ of $w_2(X)$. The Chern classes of the almost complex structure corresponding to $W$ are given by $c_1=W$ and $c_2=(W−p_1(X))/2$.

In particular any integral lift of $w_2$ of an almost-complex 3-fold, satisfying the two bottom relations is $c_1$ of some almost-complex structure. You can use the Wu formulas to give you mod $2$ relations and some others that are similar to the surface case. I think all the integral relations are derivable from this and the Hirzebruch formula (they are in the 4-dimensional analogue (maybe Rohklin is needed iirc) ). Much less is determined by the pairing on $H^3$ (then on the pairing on $H^2$) so you shouldn't expect too much more.