## Lecture Notes in Mathematics, Volume 165 |

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Page 28

[X, V A ] - Z [X,A ] . i=l i=l oo 00 j

A.) = lim tt ( V A-) ~ lim S tt (A.) = s tt (A.) i=l i=l i=l i=l j—ao j -mo Then using the

techniques of Corollary l.l0 extend to CW complexes of dimension < 2n - 2 .

[X, V A ] - Z [X,A ] . i=l i=l oo 00 j

**Proof**: \/ A. = V Ai s° f°r m < 2n - 2 i=l j=l i=l j j n ( \JA.) = lim tt ( V A-) ~ lim S tt (A.) = s tt (A.) i=l i=l i=l i=l j—ao j -mo Then using the

techniques of Corollary l.l0 extend to CW complexes of dimension < 2n - 2 .

Page 101

finite. If {X,C) is finite when X has (r-l)- cells then let Y = cone(Sn - X) have r cells.

Then {Sn+l,C) - {Y,C} - {X,C) IH"l is exact and since fX,C) and {S , C) are finite so ...

**Proof**; By induction on the number of cells of X : If X = Sn then {Sn,C) = nn(C) isfinite. If {X,C) is finite when X has (r-l)- cells then let Y = cone(Sn - X) have r cells.

Then {Sn+l,C) - {Y,C} - {X,C) IH"l is exact and since fX,C) and {S , C) are finite so ...

Page 189

Kahn for Theorem 5.l9) but the

makes it more clear what is going on. The reader may try to prove Theorem 5.l8

directly and will observe that although the idea is simple the

...

Kahn for Theorem 5.l9) but the

**proof**given here is very slick and at the same timemakes it more clear what is going on. The reader may try to prove Theorem 5.l8

directly and will observe that although the idea is simple the

**proof**is complicated...

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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