• Source: 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 path-connected space X and positive integer n there exists a group homomorphism





h

∗


:

π

n


(
X
)
→

H

n


(
X
)
,


{\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X),}


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




u

n


∈

H

n


(

S

n


)


{\displaystyle u_{n}\in H_{n}(S^{n})}

, then a homotopy class of maps



f
∈

π

n


(
X
)


{\displaystyle f\in \pi _{n}(X)}

is taken to




f

∗


(

u

n


)
∈

H

n


(
X
)


{\displaystyle f_{*}(u_{n})\in H_{n}(X)}

.
The Hurewicz theorem states cases in which the Hurewicz homomorphism is an isomorphism.

For



n
≥
2


{\displaystyle n\geq 2}

, if X is



(
n
−
1
)


{\displaystyle (n-1)}

-connected (that is:




π

i


(
X
)
=
0


{\displaystyle \pi _{i}(X)=0}

for all



i
<
n


{\displaystyle i
), then







H

i


~



(
X
)
=
0


{\displaystyle {\tilde {H_{i}}}(X)=0}

for all



i
<
n


{\displaystyle i
, and the Hurewicz map




h

∗


:

π

n


(
X
)
→

H

n


(
X
)


{\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X)}

is an isomorphism.: 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




h

∗


:

π

n
+
1


(
X
)
→

H

n
+
1


(
X
)


{\displaystyle h_{*}\colon \pi _{n+1}(X)\to H_{n+1}(X)}

is an epimorphism in this case.: 390, ? 
For



n
=
1


{\displaystyle n=1}

, the Hurewicz homomorphism induces an isomorphism







h
~




∗


:

π

1


(
X
)

/

[

π

1


(
X
)
,

π

1


(
X
)
]
→

H

1


(
X
)


{\displaystyle {\tilde {h}}_{*}\colon \pi _{1}(X)/[\pi _{1}(X),\pi _{1}(X)]\to H_{1}(X)}

, between the abelianization of the first homotopy group (the fundamental group) and the first homology group.




= Relative version

=
For any pair of spaces



(
X
,
A
)


{\displaystyle (X,A)}

and integer



k
>
1


{\displaystyle k>1}

there exists a homomorphism





h

∗


:

π

k


(
X
,
A
)
→

H

k


(
X
,
A
)


{\displaystyle h_{*}\colon \pi _{k}(X,A)\to H_{k}(X,A)}


from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both



X


{\displaystyle X}

and



A


{\displaystyle A}

are connected and the pair is



(
n
−
1
)


{\displaystyle (n-1)}

-connected then




H

k


(
X
,
A
)
=
0


{\displaystyle H_{k}(X,A)=0}

for



k
<
n


{\displaystyle k
and




H

n


(
X
,
A
)


{\displaystyle H_{n}(X,A)}

is obtained from




π

n


(
X
,
A
)


{\displaystyle \pi _{n}(X,A)}

by factoring out the action of




π

1


(
A
)


{\displaystyle \pi _{1}(A)}

. 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





π

n


(
X
,
A
)
→

π

n


(
X
∪
C
A
)
,


{\displaystyle \pi _{n}(X,A)\to \pi _{n}(X\cup CA),}


where



C
A


{\displaystyle CA}

denotes the cone of



A


{\displaystyle A}

. This statement is a special case of a homotopical excision theorem, involving induced modules for



n
>
2


{\displaystyle n>2}

(crossed modules if



n
=
2


{\displaystyle 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
)


{\displaystyle (X;A,B)}

(i.e., a space X and subspaces A, B) and integer



k
>
2


{\displaystyle k>2}

there exists a homomorphism





h

∗


:

π

k


(
X
;
A
,
B
)
→

H

k


(
X
;
A
,
B
)


{\displaystyle h_{*}\colon \pi _{k}(X;A,B)\to H_{k}(X;A,B)}


from triad homotopy groups to triad homology groups. Note that





H

k


(
X
;
A
,
B
)
≅

H

k


(
X
∪
(
C
(
A
∪
B
)
)
)
.


{\displaystyle H_{k}(X;A,B)\cong H_{k}(X\cup (C(A\cup B))).}


The Triadic Hurewicz Theorem states that if X, A, B, and



C
=
A
∩
B


{\displaystyle C=A\cap B}

are connected, the pairs



(
A
,
C
)


{\displaystyle (A,C)}

and



(
B
,
C
)


{\displaystyle (B,C)}

are



(
p
−
1
)


{\displaystyle (p-1)}

-connected and



(
q
−
1
)


{\displaystyle (q-1)}

-connected, respectively, and the triad



(
X
;
A
,
B
)


{\displaystyle (X;A,B)}

is



(
p
+
q
−
2
)


{\displaystyle (p+q-2)}

-connected, then




H

k


(
X
;
A
,
B
)
=
0


{\displaystyle H_{k}(X;A,B)=0}

for



k
<
p
+
q
−
2


{\displaystyle k
and




H

p
+
q
−
1


(
X
;
A
)


{\displaystyle H_{p+q-1}(X;A)}

is obtained from




π

p
+
q
−
1


(
X
;
A
,
B
)


{\displaystyle \pi _{p+q-1}(X;A,B)}

by factoring out the action of




π

1


(
A
∩
B
)


{\displaystyle \pi _{1}(A\cap 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




{\displaystyle \operatorname {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.




= Rational Hurewicz theorem

=
Rational Hurewicz theorem: Let X be a simply connected topological space with




π

i


(
X
)
⊗

Q

=
0


{\displaystyle \pi _{i}(X)\otimes \mathbb {Q} =0}

for



i
≤
r


{\displaystyle i\leq r}

. Then the Hurewicz map




h
⊗

Q

:

π

i


(
X
)
⊗

Q

⟶

H

i


(
X
;

Q

)


{\displaystyle h\otimes \mathbb {Q} \colon \pi _{i}(X)\otimes \mathbb {Q} \longrightarrow H_{i}(X;\mathbb {Q} )}


induces an isomorphism for



1
≤
i
≤
2
r


{\displaystyle 1\leq i\leq 2r}

and a surjection for



i
=
2
r
+
1


{\displaystyle i=2r+1}

.




Notes






References


Brown, Ronald (1989), "Triadic Van Kampen theorems and Hurewicz theorems", Algebraic topology (Evanston, IL, 1988), Contemporary Mathematics, vol. 96, Providence, RI: American Mathematical Society, pp. 39–57, doi:10.1090/conm/096/1022673, ISBN 9780821851029, MR 1022673
Brown, Ronald; Higgins, P. J. (1981), "Colimit theorems for relative homotopy groups", Journal of Pure and Applied Algebra, 22: 11–41, doi:10.1016/0022-4049(81)90080-3, ISSN 0022-4049
Brown, R.; Loday, J.-L. (1987), "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces", Proceedings of the London Mathematical Society, Third Series, 54: 176–192, CiteSeerX 10.1.1.168.1325, doi:10.1112/plms/s3-54.1.176, ISSN 0024-6115
Brown, R.; Loday, J.-L. (1987), "Van Kampen theorems for diagrams of spaces", Topology, 26 (3): 311–334, doi:10.1016/0040-9383(87)90004-8, ISSN 0040-9383
Rotman, Joseph J. (1988), An Introduction to Algebraic Topology, Graduate Texts in Mathematics, vol. 119, Springer-Verlag (published 1998-07-22), ISBN 978-0-387-96678-6
Whitehead, George W. (1978), Elements of Homotopy Theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, ISBN 978-0-387-90336-1

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