Hurewicz theorem
Encyclopedia
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 from Henri Poincaré
.
s and
homology groups.
called the Hurewicz homomorphism from the k-th homotopy group
to the k-th homology group
(with integer coefficients), which for k = 1 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 π1(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 
.
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 Hk(X,A) = 0 for k < n and Hn(X,A) is obtained from πn(X,A) by factoring out the action of π1(A). This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma.
This relative Hurewicz theorem is reformulated by 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.
from triad homotopy groups to triad homology groups. Note that Hk(X;A,B) ≅ Hk(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 Hk(X;A,B) = 0 for k < p+q−2 and Hp+q−1(X;A) is obtained from πp+q−1(X;A,B) by factoring out the action of π1(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 catn-group of an n-cube of spaces.
s satisfying the Kan condition.
for 
. Then the Hurewicz map
induces an isomorphism for
and a surjection for 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hurewicz theorem is a basic result of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, connecting homotopy theory with homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz
Witold Hurewicz
Witold Hurewicz was a Polish mathematician.- Early life and education :Witold Hurewicz was born to a Jewish family in Łódź, Russian Empire ....
, and generalizes earlier results from Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
.
Statement of the theorems
The Hurewicz theorems are a key link between homotopy groupHomotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s and
homology groups.
Absolute version
For any space X and positive integer k there exists a group homomorphismGroup homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
called the Hurewicz homomorphism from the k-th homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
to the k-th homology group
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
(with integer coefficients), which for k = 1 is equivalent to the canonical abelianization map
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group....
The Hurewicz theorem states that if X is (n − 1)-connected
N-connected
In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".-n-connected space:...
, the Hurewicz map is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
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
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
) is isomorphic to the first homology group:
The first homology group therefore vanishes if X is path-connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
and π1(X) is a perfect group
Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
.
In addition, the Hurewicz homomorphism is an epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
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 homomorphismfrom 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 Hk(X,A) = 0 for k < n and Hn(X,A) is obtained from πn(X,A) by factoring out the action of π1(A). This is proved in, for example, by induction, proving in turn the absolute version and the Homotopy Addition Lemma.
This relative Hurewicz theorem is reformulated by 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 homomorphismfrom triad homotopy groups to triad homology groups. Note that Hk(X;A,B) ≅ Hk(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 Hk(X;A,B) = 0 for k < p+q−2 and Hp+q−1(X;A) is obtained from πp+q−1(X;A,B) by factoring out the action of π1(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 catn-group of an n-cube of spaces.
Simplicial set version
The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial setSimplicial set
In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space...
s satisfying the Kan condition.
Rational Hurewicz theorem
Rational Hurewicz theorem: Let X be a simply connected topological space with for . Then the Hurewicz mapinduces an isomorphism for
and a surjection for .