Simplicial and singular homology have been applied efficaciously in several branches of mathematics; however, even for topological manifolds with mild singularities, some useful features like Poincaré duality fail to be true.
This talk is meant to be an induction to Intersection Homology (IH), which develops an effective homology theory retaining some properties of classical homology even for singular spaces.
We start with a brief recall of classical homology, stressing which theorems do not hold even in mildly singular examples. Then we see how IH is defined in analogy to the simplicial and singular case, and that a generalisation of Poincaré duality holds true.
Moving on, we revisit the definition of IH, showing it can be interpreted in a sheaf-theoretic way, leading to a perverse sheaf.
Finally, we will see some computations of IH for some low-dimensional singular manifolds, hoping to put all the theory above in context. Time permitting, as a motivation for studying IH, we could see its applications to Donaldson-Thomas theory: cohomological DT invariants are certain perverse sheaves attached to moduli spaces, called sheaves of vanishing cycles, which are indeed a manifestation of intersection cohomology.