## Lecture Notes in Mathematics, Volume 165 |

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Page 116

Then (X- Z) f Va (xvx — ^zvz) (sm) (xvx f v g > z v z) i and (tn,sm) (x -^U z) (d,d)

commute and the

2-* ZVZ) Cn tn\ m sm/ But ( #tn sm\ = tf o\ /tn sm\ /df 0 \ 9) \tn smj lo g/ Itn sm I ^0 dg

...

Then (X- Z) f Va (xvx — ^zvz) (sm) (xvx f v g > z v z) i and (tn,sm) (x -^U z) (d,d)

commute and the

**composites**are (x z) and (xvx fVg> ZVZ) (tn sm\ tn sm/ fVq (XVX2-* ZVZ) Cn tn\ m sm/ But ( #tn sm\ = tf o\ /tn sm\ /df 0 \ 9) \tn smj lo g/ Itn sm I ^0 dg

...

Page 127

m.l, and 1 i i i i X f^g^ = n>^ly • (Actually we only know tnat such maps exist with N

this case as - and - „ mean the same.) 11 m . N 1 m . -l °i fi ji Let S C - *x — *Y ...

m.l, and 1 i i i i X f^g^ = n>^ly • (Actually we only know tnat such maps exist with N

**composite**iik for some N , but we may as well replace our original m. with ID*? inthis case as - and - „ mean the same.) 11 m . N 1 m . -l °i fi ji Let S C - *x — *Y ...

Page 142

Then this leads s+t s is s-l to a spectral sequence r r Theorem 5.l: There is a

spectral sequence {E#^,d ) with El = k (W ,W ,) and d :E* - El , is the

s+t s s-l s,t s-l,t k (W ,W ,) - k , (W ,) - k , (w , ,W ) . In general, s+t s s-l s+t-l s-l s+t-l

s-l ...

Then this leads s+t s is s-l to a spectral sequence r r Theorem 5.l: There is a

spectral sequence {E#^,d ) with El = k (W ,W ,) and d :E* - El , is the

**composite**s,ts+t s s-l s,t s-l,t k (W ,W ,) - k , (W ,) - k , (w , ,W ) . In general, s+t s s-l s+t-l s-l s+t-l

s-l ...

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### Contents

Preliminaries LlBRARY | 1 |

EilenbergMacLane spaces and spectra | 31 |

SpanierWhitehead duality | 52 |

Copyright | |

3 other sections not shown

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### Common terms and phrases

abelian category abelian groups algebra Assume basepoint bouquet of spheres cancellation cofibration cohomology commutative diagram composite congruence convergent spectrum Corollary CW space define degree diagram commutes dual duality element epimorphism exists fact fibration finite CW complex finite type Freyd functor given H X,A H X/A hence Hom(H homeomorphism homology theory homotopy groups idempotent implies indecomposable induces injective integer isomorphism ker cp Let f Let G long exact sequence map f mapping cone monomorphism morphism n-connected n-l)-connected n-spheres n+l n+l n+l natural transformation non-unit non-zero observe p-torsion primes Proof prove R-module ring Schanuel's Lemma Seiten Seminar Serre Serre Spectral Sequence Sn+l Spanier spherical retracts stable homotopy strongly convergent summand theories of bidegree topology trivial weak homotopy equivalence wedge of spheres Y V B yields