ag.algebraic geometry - Computing fundamental groups and singular cohomology of projective varieties

The trick I know (learned it from Ron Livne) is to project it to some space with known homotopy / homology, throw away the ramification and branch loci to get a covering map (and you better pray it's Galois - otherwise the mess is even bigger) , and then bring the ramification back as extra relations.

e.g. here is a computation of the homotopy group of an elliptic curve E:

You have a degree 2 projection to a P¹ with four ramification points. The homotopy group of P¹ minus the four branch points is freely generated by loops about 3 of these points.

Claim: the homotopy group of E minus the ramification locus the kernel of the map from the free group on three generators: F(a,b,c) to Z / 2,
given by adding the powers on all the letters and taking mod 2 (e.g. abbac^-1b maps to 4 mod 2 = 0).

Sketch of proof: think of a,b,c, as paths in E minus the ramification points which have to glue to a closed loop, and to project to the generators of the homotopy of P¹ minus the branch points (i.e. they are "half loops" / sheet interchange about the ramification points).

Finally we have "fill" the ramification points - i.e. to bring the extra relations a², b², c². After adding these relations, our group is generated by ab, ba, ac, ca, bc, cb. Hence - since e.g. (bc)(cb) = 1 - it is generated by ab, ac, bc; hence - since (ab)(bc) = (ac) - it is generated by ab, ac. We now observe that the map which sends x to axa is the map sending an element to the inverse; which shows as that

(ab)(ca)(ab)^-1 = (ab)(ca)(ba) = (bc)^-1 ba = (cb)(ba) = ca.

Note that this is the simplest example one can give - this is a painful trick.