• There's a CST.SE thread of possible interest here: http://cstheory.stackexchange.com/questions/7958/papers-on-relation-between-computational-complexity-and-algebraic-geometry-topol

It mentions stuff like Geometric Complexity Theory, a far-out program for proving P!=NP with algebraic geometry.

• Similarly here: http://cstheory.stackexchange.com/questions/2898/applications-of-topology-to-computer-science

Mentions the thing I actually first websearched for, Herlihy's work on concurrent and distributed computing using cohomology.

• There's a CST.SE thread of possible interest here: https://cstheory.stackexchange.com/questions/7958/papers-on-relation-between-computational-complexity-and-algebraic-geometry-topol

It mentions stuff like Geometric Complexity Theory, a far-out program for proving P!=NP with algebraic geometry.

• Similarly here: https://cstheory.stackexchange.com/questions/2898/applications-of-topology-to-computer-science

Mentions the thing I actually first websearched for, Herlihy's work on concurrent and distributed computing using cohomology.