Inmathematics, theHurewicz theorem is a basic result ofalgebraic topology, connectinghomotopy theory withhomology theory via a map known as theHurewicz homomorphism. The theorem is named afterWitold Hurewicz, and generalizes earlier results ofHenri Poincaré.

The Hurewicz theorems are a key link betweenhomotopy groups andhomology groups.

For anypath-connected spaceX and positive integern there exists agroup homomorphism

${\displaystyle h_{\ast }:{\pi }_n(X)\to H_n(X),}$

called theHurewicz homomorphism, from then-thhomotopy group to then-thhomology group (with integer coefficients). It is given in the following way: choose a canonical generator${\displaystyle u_n\in H_n(S^n)}$, then a homotopy class of maps${\displaystyle f\in {\pi }_n(X)}$ is taken to${\displaystyle f_{\ast }(u_n)\in H_n(X)}$.

The Hurewicz theorem states cases in which the Hurewicz homomorphism is anisomorphism.

• For${\displaystyle n\ge 2}$, ifX is${\displaystyle (n-1)}$-connected (that is:${\displaystyle {\pi }_i(X)=0}$ for all), then${\displaystyle \widetilde{H_i}(X)=0}$ for all, and the Hurewicz map${\displaystyle h_{\ast }:{\pi }_n(X)\to H_n(X)}$ is an isomorphism.^{[1]: 366, Thm.4.32} This implies, in particular, that thehomological connectivity equals thehomotopical connectivity when the latter is at least 1. In addition, the Hurewicz map${\displaystyle h_{\ast }:{\pi }_{n+1}(X)\to H_{n+1}(X)}$ is anepimorphism in this case.^{[1]: 390, ?}
• For${\displaystyle n=1}$, the Hurewicz homomorphism induces anisomorphism${\displaystyle {\tilde{h}}_{\ast }:{\pi }_1(X)/[{\pi }_1(X),{\pi }_1(X)]\to H_1(X)}$, between theabelianization of the first homotopy group (thefundamental group) and the first homology group.

For anypair of spaces${\displaystyle (X,A)}$ and integer${\displaystyle k>1}$ there exists a homomorphism

${\displaystyle h_{\ast }:{\pi }_k(X,A)\to H_k(X,A)}$

from relative homotopy groups to relative homology groups. The Relative Hurewicz Theorem states that if both${\displaystyle X}$ and${\displaystyle A}$ are connected and the pair is${\displaystyle (n-1)}$-connected then${\displaystyle H_k(X,A)=0}$ for and${\displaystyle H_n(X,A)}$ is obtained from${\displaystyle {\pi }_n(X,A)}$ by factoring out the action of${\displaystyle {\pi }_1(A)}$. 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 byBrown & Higgins (1981) as a statement about the morphism

${\displaystyle {\pi }_n(X,A)\to {\pi }_n(X\cup CA),}$

where${\displaystyle CA}$ denotes thecone of${\displaystyle A}$. This statement is a special case of ahomotopical excision theorem, involving induced modules for${\displaystyle n>2}$ (crossed modules if${\displaystyle n=2}$), which itself is deduced from a higher homotopyvan Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

For any triad of spaces${\displaystyle (X;A,B)}$ (i.e., a spaceX and subspacesA,B) and integer${\displaystyle k>2}$ there exists a homomorphism

${\displaystyle h_{\ast }:{\pi }_k(X;A,B)\to H_k(X;A,B)}$

from triad homotopy groups to triad homology groups. Note that

${\displaystyle H_k(X;A,B)\cong H_k(X\cup (C(A\cup B))).}$

The Triadic Hurewicz Theorem states that ifX,A,B, and${\displaystyle C=A\cap B}$ are connected, the pairs${\displaystyle (A,C)}$ and${\displaystyle (B,C)}$ are${\displaystyle (p-1)}$-connected and${\displaystyle (q-1)}$-connected, respectively, and the triad${\displaystyle (X;A,B)}$ is${\displaystyle (p+q-2)}$-connected, then${\displaystyle H_k(X;A,B)=0}$ for and${\displaystyle H_{p+q-1}(X;A)}$ is obtained from${\displaystyle {\pi }_{p+q-1}(X;A,B)}$ by factoring out the action of${\displaystyle {\pi }_1(A\cap B)}$ 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${\displaystyle {\mathrm{cat}}^n}$-group of ann-cube of spaces.

The Hurewicz theorem for topological spaces can also be stated forn-connectedsimplicial sets satisfying the Kan condition.^[2]

Rational Hurewicz theorem:^[3][4] LetX be a simply connected topological space with${\displaystyle {\pi }_i(X)\otimes \mathbb{Q}=0}$ for${\displaystyle i\le r}$. Then the Hurewicz map

${\displaystyle h\otimes \mathbb{Q}:{\pi }_i(X)\otimes \mathbb{Q}⟶H_i(X;\mathbb{Q})}$

induces an isomorphism for${\displaystyle 1\le i\le 2r}$ and a surjection for${\displaystyle i=2r+1}$.

• 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

• Theorems in homotopy theory
• Homology theory
• Theorems in algebraic topology

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