One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.[1]
If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism.
The Eilenberg–Steenrod axioms apply to a sequence of functors from the category of pairs of topological spaces to the category of abelian groups, together with a natural transformation called the boundary map (here is a shorthand for . The axioms are:
Homotopy: Homotopic maps induce the same map in homology. That is, if is homotopic to , then their induced homomorphisms are the same.
Excision: If is a pair and U is a subset of A such that the closure of U is contained in the interior of A, then the inclusion map induces an isomorphism in homology.
Dimension: Let P be the one-point space; then for all .
Additivity: If , the disjoint union of a family of topological spaces , then
Exactness: Each pair (X, A) induces a long exact sequence in homology, via the inclusions and :
If P is the one point space, then is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.
Consequences
Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.
The homology of some relatively simple spaces, such as n-spheres, can be calculated directly from the axioms. From this it can be easily shown that the (n − 1)-sphere is not a retract of the n-disk. This is used in a proof of the Brouwer fixed point theorem.
Bredon, Glen (1993). Topology and Geometry. Graduate Texts in Mathematics. Vol. 139. New York: Springer-Verlag. doi:10.1007/978-1-4757-6848-0. ISBN0-387-97926-3. MR 1224675.
January 24, 2023
eilenberg, steenrod, axioms, mathematics, specifically, algebraic, topology, properties, that, homology, theories, topological, spaces, have, common, quintessential, example, homology, theory, satisfying, axioms, singular, homology, developed, samuel, eilenber. In mathematics specifically in algebraic topology the Eilenberg Steenrod axioms are properties that homology theories of topological spaces have in common The quintessential example of a homology theory satisfying the axioms is singular homology developed by Samuel Eilenberg and Norman Steenrod One can define a homology theory as a sequence of functors satisfying the Eilenberg Steenrod axioms The axiomatic approach which was developed in 1945 allows one to prove results such as the Mayer Vietoris sequence that are common to all homology theories satisfying the axioms 1 If one omits the dimension axiom described below then the remaining axioms define what is called an extraordinary homology theory Extraordinary cohomology theories first arose in K theory and cobordism Contents 1 Formal definition 2 Consequences 3 Dimension axiom 4 See also 5 Notes 6 ReferencesFormal definition EditThe Eilenberg Steenrod axioms apply to a sequence of functors H n displaystyle H n from the category of pairs X A displaystyle X A of topological spaces to the category of abelian groups together with a natural transformation H i X A H i 1 A displaystyle partial colon H i X A to H i 1 A called the boundary map here H i 1 A displaystyle H i 1 A is a shorthand for H i 1 A displaystyle H i 1 A emptyset The axioms are Homotopy Homotopic maps induce the same map in homology That is if g X A Y B displaystyle g colon X A rightarrow Y B is homotopic to h X A Y B displaystyle h colon X A rightarrow Y B then their induced homomorphisms are the same Excision If X A displaystyle X A is a pair and U is a subset of A such that the closure of U is contained in the interior of A then the inclusion map i X U A U X A displaystyle i colon X setminus U A setminus U to X A induces an isomorphism in homology Dimension Let P be the one point space then H n P 0 displaystyle H n P 0 for all n 0 displaystyle n neq 0 Additivity If X a X a displaystyle X coprod alpha X alpha the disjoint union of a family of topological spaces X a displaystyle X alpha then H n X a H n X a displaystyle H n X cong bigoplus alpha H n X alpha Exactness Each pair X A induces a long exact sequence in homology via the inclusions i A X displaystyle i colon A to X and j X X A displaystyle j colon X to X A H n A i H n X j H n X A H n 1 A displaystyle cdots to H n A xrightarrow i H n X xrightarrow j H n X A xrightarrow partial H n 1 A to cdots dd If P is the one point space then H 0 P displaystyle H 0 P is called the coefficient group For example singular homology taken with integer coefficients as is most common has as coefficients the integers Consequences EditSome facts about homology groups can be derived directly from the axioms such as the fact that homotopically equivalent spaces have isomorphic homology groups The homology of some relatively simple spaces such as n spheres can be calculated directly from the axioms From this it can be easily shown that the n 1 sphere is not a retract of the n disk This is used in a proof of the Brouwer fixed point theorem Dimension axiom EditA homology like theory satisfying all of the Eilenberg Steenrod axioms except the dimension axiom is called an extraordinary homology theory dually extraordinary cohomology theory Important examples of these were found in the 1950s such as topological K theory and cobordism theory which are extraordinary cohomology theories and come with homology theories dual to them See also EditZig zag lemmaNotes Edit http www math uiuc edu K theory 0245 survey pdf bare URL PDF References EditEilenberg Samuel Steenrod Norman E 1945 Axiomatic approach to homology theory Proceedings of the National Academy of Sciences of the United States of America 31 4 117 120 Bibcode 1945PNAS 31 117E doi 10 1073 pnas 31 4 117 MR 0012228 PMC 1078770 PMID 16578143 Eilenberg Samuel Steenrod Norman E 1952 Foundations of algebraic topology Princeton New Jersey Princeton University Press MR 0050886 Bredon Glen 1993 Topology and Geometry Graduate Texts in Mathematics Vol 139 New York Springer Verlag doi 10 1007 978 1 4757 6848 0 ISBN 0 387 97926 3 MR 1224675 Retrieved from https en wikipedia org w index php title Eilenberg Steenrod axioms amp oldid 1092646631, wikipedia, wiki, book, books, library,
