## Lecture Notes in Mathematics, Volume 165 |

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Page 22

Y ; we wish to construct a homotopy H:X x I - Y with H(x,0) = f ( x) , H( x, l) = * . X is

obtained from * by adjoining various ... Now define H||n as follows: write l o l 1 en

=

Y ; we wish to construct a homotopy H:X x I - Y with H(x,0) = f ( x) , H( x, l) = * . X is

obtained from * by adjoining various ... Now define H||n as follows: write l o l 1 en

=

**Sn**-**l**A [0,l) where [0,l) has base point 0 . For (x,t) € let H((x,t),s) = < S((x,t),^) 0 ...Page 23

If Q:

A U e° and n (Y.B) - 0 , then any map 9 n f:(X,A) -• (Y,B) is homotopic rel. A to

some f ' where f(X) c B . Proof: Let i:(En,Sn - (X,A) be the obvious map where i|En

...

If Q:

**Sn l**-• A then C will often be 8 n rcitten as A U e . Observe e Lemma l.l2: If X =A U e° and n (Y.B) - 0 , then any map 9 n f:(X,A) -• (Y,B) is homotopic rel. A to

some f ' where f(X) c B . Proof: Let i:(En,Sn - (X,A) be the obvious map where i|En

...

Page 28

[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 ( \J

A.) = lim tt ( V A-) ~ lim S tt (A.) = s tt (A.) ... For 00 00 example if X = V Sn , then [l]

€ [X,X] but l ^ E [X,

[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 ( \J

A.) = lim tt ( V A-) ~ lim S tt (A.) = s tt (A.) ... For 00 00 example if X = V Sn , then [l]

€ [X,X] but l ^ E [X,

**Sn**] .**l**X X l Theorem l.2l (Generalized Freudenthal) : If X is ...### 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