现代几何方法和应用（第3卷）

Inexpositionsoftheelementsoftopologyitiscustomaryforhomologytobegivenafundamentalrole.SincePoincare,wholaidthefoundationsoftopology,homologytheoryhasbeenregardedastheappropriateprimarybasisforanintroductiontothemethodsofalgebraictopology.Fromhomotopytheory,ontheotherhand,onlythefundamentalgroupandcovering-spacetheoryhavetraditionallybeenincludedamongthebasicinitialconcepts.Essentiallyallelementaryclassicaltextbooksoftopology（thebestofwhichis,intheopinionofthepresentauthors,SeifertandThrelfall'sATextbookofTopology）beginwiththehomologytheoryofoneoranotherclassofcomplexes.Onlyatalaterstage（andthenstillfromahomologicalpointofview）dofibre-spacetheoryandthegeneralproblemofclassifyinghomotopyclassesofmaps（homotopytheory）comeinforconsideration.However,methodsdevelopedininvestigatingthetopologyofdifferentiablemanifolds,andintensivelyelaboratedfromthe1930sonwards（byWhitneyandothers）,nowpermitawholesalereorganizationofthestandardexpositionOfthefundamentalsofmoderntopology.Inthisnewapproach,whichresemblesmorethatofclassicalanalysis,thesefundamentalsturnouttoconsistprimarilyoftheelementarytheoryofsmoothmanifolds,homotopytheorybasedonthese,andsmoothfibrespaces.Furthermore,overthedecadeofthe1970sitbecameclearthatexactlythiscomplexoftopologicalideasandmethodswereprovingtobefundamentallyapplicableinvariousareasofmodernphysics.