Gives a homomorphism from homotopy groups to homology groups
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 path-connected space X and positive integer n there exists a group homomorphism
![{\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/421f612c974971420245e0aeec1bb80a1de443ce)
called the Hurewicz homomorphism, from the n-th homotopy group to the n-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator
, then a homotopy class of maps
is taken to
.
The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.
- For
, if X is
-connected (that is:
for all
), then
for all
, and the Hurewicz map
is an isomorphism.[1]: 366, Thm.4.32 This implies, in particular, that the homological connectivity equals the homotopical connectivity when the latter is at least 1. In addition, the Hurewicz map
is an epimorphism in this case.[1]: 390, ?
- For
, the Hurewicz homomorphism induces an isomorphism
, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.
Relative version
For any pair of spaces
and integer
there exists a homomorphism
![{\displaystyle h_{*}\colon \pi _{k}(X,A)\to H_{k}(X,A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4842da11bba3b281ee3c3cb7f9da4c29fc627c)
from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both
and
are connected and the pair is
-connected then
for
and
is obtained from
by factoring out the action of
. This is proved in, for example, Whitehead (1978) by induction, proving in turn the absolute version and the Homotopy Addition Lemma.
This relative Hurewicz theorem is reformulated by Brown & Higgins (1981) as a statement about the morphism
![{\displaystyle \pi _{n}(X,A)\to \pi _{n}(X\cup CA),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e70bf5f7a631e438c51d5b8d57a561df953d1aa8)
where
denotes the cone of
. This statement is a special case of a homotopical excision theorem, involving induced modules for
(crossed modules if
), 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
(i.e., a space X and subspaces A, B) and integer
there exists a homomorphism
![{\displaystyle h_{*}\colon \pi _{k}(X;A,B)\to H_{k}(X;A,B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1da2e15fe21252123843e5dde7f9c367598960)
from triad homotopy groups to triad homology groups. Note that
![{\displaystyle H_{k}(X;A,B)\cong H_{k}(X\cup (C(A\cup B))).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b868e08f3e9cdb81fa40d8256f17be5b58955db3)
The Triadic Hurewicz Theorem states that if X, A, B, and
are connected, the pairs
and
are
-connected and
-connected, respectively, and the triad
is
-connected, then
for
and
is obtained from
by factoring out the action of
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
-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.[2]
Rational Hurewicz theorem
Rational Hurewicz theorem:[3][4] Let X be a simply connected topological space with
for
. Then the Hurewicz map
![{\displaystyle h\otimes \mathbb {Q} \colon \pi _{i}(X)\otimes \mathbb {Q} \longrightarrow H_{i}(X;\mathbb {Q} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675fab2a1d98ed8a3d1e3d5f9bb178bf87b8c058)
induces an isomorphism for
and a surjection for
.