Primary operations

Abstract

It is good general philosophy that if you want to show that a geometrical construction is possible, you go ahead and perform it; but if you want to show that a proposed geometric construction is impossible, you have to find a topological invariant which shows the impossibility. Among topological invariants we meet first the homology and cohomology groups, with their additive and multiplicative structure. Afte that we meet cohomology operations, such as the celebrated Steenrod square. I recall that this is a homomorphism

$$s{q^i}:{H^n}\left( {x,y;{z_2}} \right) \to {H^{n + 1}}\left( {x,y;{z_2}} \right)$$

defined for each pair (X,Y) and for all non-negative integers i and n. (Hⁿ is to be interpreted as singular cohomology.) The Steenrod square enjoys the following properties: