On manifolds, ordinary (co)homology has a very important property:
Poincaré duality. It allows e.g. to define the signature, a powerful
invariant. On singular spaces unfortunately, Poincaré duality does not
hold in general for ordinary (singular) (co)homology. To get a
(co)homology theory satisfying Poincaré duality one has to give up some
of the Eilenberg-Steenrod axioms.
In the talk I want to give an introduction to two examples for such
(co)homology theories: Intersection (Co)homology and Intersection Space
(Co)homology.
While one defines Intersection (Co)homology by allowing only (co)chains
that satisfy certain properties related to the singular sets and hence
defining a subcomplex of the usual (co)chain complex one gets
Intersection Space (Co)homology by assigning a new CW complex to the
singular space and then taking its usual (co)homology. An alternative
description of the latter using differential forms will also be given.