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1、Exactsequencesareusedallthetimeinalgebraictopology,butinmanysituationstoocomplicatedtobedescribedbyanexactsequencethereisaconsiderablymoresubtleandsophisticatedalgebraicmachineavailable,calledaspectralsequence.Forexample,thehomologyorcohomologylongexactsequenceofapairX;Agener
2、-alizestoaspectralsequenceassociatedtoanarbitraryincreasingsequencesofsub-SspacesX0X1XwithXiXi.Similarly,theMayer-VietorissequenceforadecompositionXA[BgeneralizestoaspectralsequenceassociatedtoacoverofXbyanynumberofsets.Withthisgreatincreaseingeneralitycomes,notsurprisi
3、ngly,acorrespondingincreaseincomplexity.Thiscanbeaseriousobstacletounderstandingspectralse-quencesonfirstexposure.Butoncetheinitialhurdleof‘believingin’spectralse-quencesissurmounted,onecannothelpbutbeamazedattheirpower.1.1TheHomologySpectralSequenceOnecanthinkofaspectralseque
4、nceasabookconsistingofasequenceofpages,eachofwhichisatwo-dimensionalarrayofabeliangroups.Oneachpagetherearemapsbetweenthegroups,andthesemapsformchaincomplexes.Thehomologygroupsofthesechaincomplexesarepreciselythegroupswhichappearonthenextpage.Forexample,intheSerrespectralsequ
5、enceforhomologythefirstfewpageshavetheformshowninthefigurebelow,whereeachdotrepresentsagroup.312Onlythefirstquadrantofeachpageisshownbecauseoutsidethefirstquadrantallthegroupsarezero.Themapsformingchaincomplexesoneachpageareknownas2Chapter1TheSerreSpectralSequencedifferentials.On
6、thefirstpagetheygooneunittotheleft,onthesecondpagetwounitstotheleftandoneunitup,onthethirdpagethreeunitstotheleftandtwounitsup,andingeneralontherthpagetheygorunitstotheleftandr−1unitsup.Ifonefocusesonthegroupatthep;qlatticepointineachpage,forfixedpandq,thenasonekeepsturningtosu
7、ccessivepages,thedifferentialsenteringandleavingthisp;qgroupwilleventuallybezerosincetheywilleithercomefromorgotogroupsoutsidethefirstquadrant.Hence,passingtothenextpagebycomputinghomologyatthep;qspotwithrespecttothesedifferentialswillnotchangethep;qgroup.Sinceeachp;qgroupeven
8、tuallystabilizesinthisway,thereisawell-definedlimitingpageforthespect
