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 from Henri Poincaré.

tatement of the theorems

The Hurewicz theorems are a key link between homotopy groups and
homology groups.

Absolute version

For any space "X" and positive integer "k" there exists a group homomorphism

:$h\_\{ast\}colon,\; pi\_k(X)\; o\; H\_k(X)\; ,!$

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

:$h\_\{ast\}colon,\; pi\_1(X)\; o\; pi\_1(X)/\; [\; pi\_1(X),\; pi\_1(X)]\; .\; ,!$

The Hurewicz theorem states that if "X" is ("n"−1)-connected, the Hurewicz map is an isomorphism for all "k" ≤ "n". In particular, this theorem says that the abelianization of the first homotopy group (the fundamental group) is isomorphic to the first homology group:

:$H\_1(X)\; cong\; pi\_1(X)/\; [\; pi\_1(X),\; pi\_1(X)]\; .\; ,!$

The first homology group therefore vanishes if "X" is path-connected and π₁("X") is a perfect group.

Relative version

For any pair of spaces ("X","A") and integer "k" > 1 there exists a homomorphism

:$h\_\{ast\}colon\; pi\_k(X,A)\; o\; H\_k(X,A)\; ,!$

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 "H"_"k"("X","A") = 0 for "k" < "n" and "H"_"n"("X","A") is obtained from π_"n"("X","A") by factoring out the action of π₁("A"). This is proved in, for example, Harvtxt|Whitehead|1978 by induction, proving in turn the absolute version and the Homotopy Addition Lemma.

This relative Hurewicz theorem is reformulated by Harvtxt|Brown|Higgins|1981 as a statement about the morphism :$pi\_n(X,A)\; o\; pi\_n(X\; cup\; CA)\; ,!.$

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 homomorphism

:$h\_\{ast\}colon\; pi\_k(X;A,B)\; o\; H\_k(X;A,B)\; ,!$

from triad homotopy groups to triad homology groups. Note that "H"_"k"("X";"A","B") ≅ "H"_"k"("X"∪("C"("A"∪"B")). 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 "H"_"k"("X";"A","B") = 0 for "k" < "p"+"q"−2 and "H"_"p"+"q"−1("X";"A") is obtained from π_"p"+"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 cat^"n"-group of an "n"-cube of spaces.

References

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