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1、Therearemanysituationsinalgebraictopologywheretherelationshipbetweencertainhomotopy,homology,orcohomologygroupsisexpressedperfectlybyanexactsequence.Inothercases,however,therelationshipmaybemorecomplicatedandamorepowerfulalgebraictoolisneeded.Inawidevarietyofsituationsspectralsequ
2、encesprovidesuchatool.Forexample,insteadofconsideringjustapairX;Aandtheassociatedlongexactsequencesofhomologyandcohomologygroups,onecouldconsideranarbitraryincreasingsequenceofsubspacesX0X1XSwithXiXi,andthenthereareassociatedhomologyandcohomologyspectralsequences.Similarly,t
3、heMayer-VietorissequenceforadecompositionXA[BgeneralizestoaspectralsequenceassociatedtoacoverofXbyanynumberofsets.Withthisgreatincreaseingeneralitycomes,notsurprisingly,acorrespondingincreaseincomplexity.Thiscanbeaseriousobstacletounderstandingspectralse-quencesonfirstexposure.Buto
4、ncetheinitialhurdleof‘believingin’spectralse-quencesissurmounted,onecannothelpbutbeamazedattheirpower.1.1TheHomologySpectralSequenceOnecanthinkofaspectralsequenceasabookconsistingofasequenceofpages,eachofwhichisatwo-dimensionalarrayofabeliangroups.Oneachpagetherearemapsbetweentheg
5、roups,andthesemapsformchaincomplexes.Thehomologygroupsofthesechaincomplexesarepreciselythegroupswhichappearonthenextpage.Forexample,intheSerrespectralsequenceforhomologythefirstfewpageshavetheformshowninthefigurebelow,whereeachdotrepresentsagroup.312Onlythefirstquadrantofeachpageissh
6、ownbecauseoutsidethefirstquadrantallthegroupsarezero.Themapsformingchaincomplexesoneachpageareknownas2Chapter1TheSerreSpectralSequencedifferentials.Onthefirstpagetheygooneunittotheleft,onthesecondpagetwounitstotheleftandoneunitup,onthethirdpagethreeunitstotheleftandtwounitsup,anding
7、eneralontherthpagetheygorunitstotheleftandr−1unitsup.Ifonefocusesonthegroupatthep;qlatticepointineachpage,forfixedpandq,thenasonekeepsturningtosuccessivepages,thedifferentialsenteringandleavingthisp;qgroupwilleventuallybezerosincetheywilleithercomefromorgotogroupsoutsidethefirstquad
8、rant.Hence,passingtothenextpageby
