Hurewicz theorem

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

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<n} ), then H i ~ ( X ) = 0 {\displaystyle {\tilde {H_{i}}}(X)=0} for all i < n {\displaystyle i<n} , and the Hurewicz map h ∗ : π n ( X ) → H n ( X ) {\displaystyle h_{*}\colon \pi _{n}(X)\to H_{n}(X)} 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 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.[1]: 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<n} 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<p+q-2} 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.[2]

Rational Hurewicz theorem

Rational Hurewicz theorem:[3][4] 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

  1. ^ a b Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79160-1
  2. ^ Goerss, Paul G.; Jardine, John Frederick (1999), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-7643-6064-1, III.3.6, 3.7
  3. ^ Klaus, Stephan; Kreck, Matthias (2004), "A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres", Mathematical Proceedings of the Cambridge Philosophical Society, 136 (3): 617–623, Bibcode:2004MPCPS.136..617K, doi:10.1017/s0305004103007114, S2CID 119824771
  4. ^ Cartan, Henri; Serre, Jean-Pierre (1952), "Espaces fibrés et groupes d'homotopie, II, Applications", Comptes rendus de l'Académie des Sciences, 2 (34): 393–395

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