## Lecture Notes in Mathematics, Volume 165 |

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Page 96

Because of these difficulties, the closest we can hope to come in getting a good

description of equivalence classes of spaces will necessitate weakening the

equivalence: write X = Y,X

YVZ .

Because of these difficulties, the closest we can hope to come in getting a good

description of equivalence classes of spaces will necessitate weakening the

equivalence: write X = Y,X

**congruent**to Y , if and only if for some space Z , XVZ-YVZ .

Page 128

4.39 , X' i£ q q P q P a

retract P P of X' . Hence X' □ Y' and thus X = Y . So X V B = Vx = Vy

P P P P P P P Y v B so X □ Y and the Proposition is proved. n-l Apply Theorem

4.40 to a space X ...

4.39 , X' i£ q q P q P a

**congruence**retract of Y' . Similarly Y' is a**congruence**retract P P of X' . Hence X' □ Y' and thus X = Y . So X V B = Vx = Vy

P P P P P P P Y v B so X □ Y and the Proposition is proved. n-l Apply Theorem

4.40 to a space X ...

Page 130

Th by the uniqueness part of the decomposition theorem, = Y^ . Th is, X VBV V Bx

= X' V B' V Bx, . By the dependence of p q/p q p q/p q spherical retracts on the

Th by the uniqueness part of the decomposition theorem, = Y^ . Th is, X VBV V Bx

= X' V B' V Bx, . By the dependence of p q/p q p q/p q spherical retracts on the

**congruence**class, we get X = X' since P P X and X' have trivial spherical retract.### What people are saying - Write a review

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### Contents

Preliminaries LlBRARY | 1 |

EilenbergMacLane spaces and spectra | 31 |

SpanierWhitehead duality | 52 |

Copyright | |

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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