Hurewicz theorem

In mathematics, the Hurewicz theorem is an important idea in a part of math called algebraic topology. This area helps us understand shapes by looking at how they can be stretched and bent. The Hurewicz theorem connects two ways of studying shapes: homotopy theory and homology theory. It does this using a special map called the Hurewicz homomorphism.

The theorem is named after Witold Hurewicz, a mathematician who worked on these ideas. His work built on earlier discoveries by another famous mathematician, Henri Poincaré. The Hurewicz theorem helps mathematicians see deeper relationships between different ways of describing shapes, making it a key part of how we study space and form in advanced math.

Statement of the theorems

The Hurewicz theorems connect two important ideas in mathematics: homotopy groups and homology groups.

For any connected space and a positive whole number, there is a special mapping called the Hurewicz homomorphism. This mapping links the n-th homotopy group to the n-th homology group.

The Hurewicz theorem tells us when this mapping works perfectly as a matching pair. For numbers greater than or equal to 2, if the space has certain simple properties, the mapping is perfect. For the number 1, the theorem shows a matching between a simplified version of the first homotopy group and the first homology group.