## Lecture Notes in Mathematics, Volume 165 |

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Page 9

Definition: p:E - B is a fibre map if and only if p satisfies the covering homotopy

property: if W is any space and W x O f P H w x I -> B is a

then there exists G:W x I - E making the diagram commute . If p is a fibre map and

F ...

Definition: p:E - B is a fibre map if and only if p satisfies the covering homotopy

property: if W is any space and W x O f P H w x I -> B is a

**commutative diagram**,then there exists G:W x I - E making the diagram commute . If p is a fibre map and

F ...

Page 105

Thus l 2 Proof: The

C fl+f - C. , , with cof ibre SC/1% by Property 5) . 2 <Vf2> (J) But C = X . If slml +

s2m2 = l ' tnen the

yields ...

Thus l 2 Proof: The

**commutative diagram**0 (frf2) xvx — - — - — *. z yields a mapC fl+f - C. , , with cof ibre SC/1% by Property 5) . 2 <Vf2> (J) But C = X . If slml +

s2m2 = l ' tnen the

**commutative diagram**1l1 xvx lfrf2) -> z t s m 2 21 fl + f2 Zyields ...

Page 110

X is an Ftt#

homology isomorphism (Fn^(B) - H^(B)) . Thus the diagram differs from being

commutative only by the torsion in {B,X) , which is finitely generated. So choose t

...

X is an Ftt#

**commutative diagram**. g is a homotopy equivalence since it induces ahomology isomorphism (Fn^(B) - H^(B)) . Thus the diagram differs from being

commutative only by the torsion in {B,X) , which is finitely generated. So choose t

...

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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