# Hurewicz theorem

In mathematics, the **Hurewicz theorem** is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the **Hurewicz homomorphism**. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

## Statement of the theorems

The Hurewicz theorems are a key link between homotopy groups and homology groups.

### Absolute version

For any space *X* and positive integer *k* there exists a group homomorphism

called the Hurewicz homomorphism from the *k*-th homotopy group to the *k*-th homology group (with integer coefficients), which for *k* = 1 and *X* path-connected is equivalent to the canonical abelianization map

The Hurewicz theorem states that if *X* is (*n* − 1)-connected, the Hurewicz map is an isomorphism for all *k* ≤ *n* when *n* ≥ *2* and abelianization for *n* = *1*. In particular, this theorem says that the abelianization of the first homotopy group (the fundamental group) is isomorphic to the first homology group:

The first homology group therefore vanishes if *X* is path-connected and π_{1}(*X*) is a perfect group.

In addition, the Hurewicz homomorphism is an epimorphism from whenever X is (*n* − 1)-connected, for .

The group homomorphism is given in the following way. Choose canonical generators . Then a homotopy class of maps is taken to .

### Relative version

For any pair of spaces (*X*,*A*) and integer *k* > 1 there exists a homomorphism

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if each of *X*, *A* are connected and the pair (*X*,*A*) is (*n*−1)-connected then *H*_{k}(*X*,*A*) = 0 for *k* < *n* and *H*_{n}(*X*,*A*) is obtained from π_{n}(*X*,*A*) by factoring out the action of π_{1}(*A*). This is proved in, for example, Template:Harvtxt by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by Template:Harvtxt as a statement about the morphism

This statement is a special case of a homotopical excision theorem, involving induced modules for n>2 (crossed modules if n=2), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

### Triadic version

For any triad of spaces (*X*;*A*,*B*) (i.e. space *X* and subspaces *A*,*B*) and integer *k* > 2 there exists a homomorphism

from triad homotopy groups to triad homology groups. Note that *H*_{k}(*X*;*A*,*B*) ≅ *H*_{k}(*X*∪(*C*(*A*∪*B*)). The Triadic Hurewicz Theorem states that if *X*, *A*, *B*, and *C* = *A*∩*B* are connected, the pairs (*A*,*C*), (*B*,*C*) are respectively (*p*−1)-, (*q*−1)-connected, and the triad (*X*;*A*,*B*) is *p*+*q*−2 connected, then *H*_{k}(*X*;*A*,*B*) = 0 for *k* < *p*+*q*−2 and *H*_{p+q−1}(*X*;*A*) is obtained from π_{p+q−1}(*X*;*A*,*B*) by factoring out the action of π_{1}(*A*∩*B*) and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental cat^{n}-group of an *n*-cube of spaces.

### Simplicial set version

The Hurewicz theorem for topological spaces can also be stated for *n*-connected simplicial sets satisfying the Kan condition.^{[1]}

### Rational Hurewicz theorem

**Rational Hurewicz theorem: ^{[2]}^{[3]}** Let

*X*be a simply connected topological space with for . Then the Hurewicz map

induces an isomorphism for and a surjection for .

## References

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