| contents. i ntroduction introductiontotheenglishtranslation chapter1thesimplesttopologicalproperties chapter2topologicalspacesfibrationshomotopies 1observationsfromgeneraltopologyterminology 2 homotopieshomotopytype 3 coveringhomotopiesfibrations 4 homotopygroupsan({fibrationsexactsequencesexamples chapter3simplicialcomplexesandcw-complexeshomologyand cohomologytheirrelationtohomotopytheoryobstructions 1 simplicialcomplexes 2 thehomologyandcohomologygroupspoincareduality 3 relativehomologytheexactsequenceofapairaxiomsfor homologytheorycw-complexes 4 simplicialcomplexesandotherhomologytheoriessingular homologycoveringsandsheavestheexactsequenceofsheaves andcohomology 5 homologytheoryofnon-simply-connectedspacescomplexes ofmodulesreidemeistertorsionsimplehomotopytype .6 simplicialandcellbundleswithastructuregroupobstructions universalobjects:universalfiberbundlesandtheuniversal propertyofeilenberg-maclanecomplexescohomology operationsthesteenrodalgebratheadamsspectralsequence 7 theclassicalapparatusofhomotopytheorytheleray spectralsequencethehomologytheoryoffiberbundlesthe cartan-serremethodthepostnikovtowertheadamsspectral sequence 8 definitionandpropertiesofk-theorytheatiyah-hirzebruch spectralsequenceadamsoperationsanaloguesofthethom isomorphismandtheriemann-rochtheoremellipticoperators andk-theorytransformationgroupsfour-dimensional manifolds 9 bordismandcobordismtheoryasgeneralizedhomology andcohomologycohomologyoperationsincobordismthe adams-novikovspectralsequenceformalgroupsactionsof cyclicgroupsandthecircleonmanifolds chapter4smoothmanifolds 1 basicconceptssmoothfiberbundlesconnexionscharacteristic classes 2 thehomologytheoryofsmoothmanifoldscomplex manifoldstheclassicalglobalcalculusofvariationsh-spaces multi-valuedfunctionsandfunctionals.. 3 smoothmanifoldsandhomotopytheoryframedmanifolds bordismsthomspacesthehirzebruchformulaeestimatesof theordersofhomotopygroupsofspheresmilnor'sexample theintegralpropertiesofcobordisms 4 classificationproblemsinthetheoryofsmoothmanifoldsthe theoryofimmersionsmanifoldswiththehomotopytypeof asphererelationshipsbetweensmoothandpl-manifolds integralpontryaginclasses 5 theroleofthefundamentalgroupintopologymanifoldsof lowdimension(n=2,3)knotstheboundaryofanopen manifoldthetopologicalinvarianceoftherationalpontryagin classestheclassificationtheoryofnon-simply-connected manifoldsofdimension5highersignatureshermitian k-theorygeometrictopology:theconstructionofnon-smooth homeomorphismsmilnor'sexampletheannulusconjecture topologicalandpl-structures concludingremarks appendixrecentdevelopmentsinthetopologyof3-manifolds andknots 1 introduction:recentdevelopmentsintopology 2 knots:theclassicalandmodernapproachestothealexander polynomialjones-typepolynomials 3 vassilievinvariants 4 newtopologicalinvariantsfor3-manifoldstopological quantumfieldtheories bibliography index... |
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