Introduction to Homotopy Theory (Universitext)

By Martin Arkowitz

The unifying subject of this book is the Eckmann-Hilton duality concept, to not be discovered because the motif of the other text.  considering many subject matters happen in twin pairs, this gives motivation for the tips and decreases the volume of repetitious fabric. This conscientiously written textual content strikes at a steady speed, in spite of rather complicated fabric. additionally, there's a wealth of illustrations and workouts. The more challenging routines are starred, and tricks to them are given on the finish of the book.
Key issues include:
*basic homotopy
*H-Spaces and Co-H-Spaces;
*cofibrations and fibrations;
*exact sequences;
*applications of exactness;
*homotopy pushouts and pullbacks and
 the classical theorems of homotopy theory;
*homotopy and homology decompositions;
*homotopy units; and
*obstruction theory.
The booklet is written as a textual content for a moment path in algebraic topology, for a issues seminar in homotopy concept, or for self guide.

Show description

Quick preview of Introduction to Homotopy Theory (Universitext) PDF

Show sample text content

Eight. 1 creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. 2 The Set rX, Y s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. three type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. four Loop and workforce constitution in rX, Y s . . . . . . . . . . . . . . . . . . . . . . routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 267 267 270 275 279 nine Obstruction conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine. 1 advent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine. 2 Obstructions utilizing Homotopy Decompositions .

2 three. three Fibrations such a lot of our dialogue of fibrations is simply an easy dualization of the former part on cofibrations. For this fabric, we nation those effects and sometimes point out an evidence, yet our therapy is sketchy. even if, we supply info for the cloth on fibrations that aren't duals of ends up in part three. 2. we start with a few definitions. Definition three. three. 1 A map p : E Ñ B has the homotopy lifting estate or the overlaying homotopy estate with appreciate to an area W if for maps g0 : W Ñ B and h0 : W Ñ E and homotopy gt : W Ñ B of g0 such that ph0 ✏ g0 , there exists a homotopy ht : W Ñ E of h0 such that pht ✏ gt .

Four. nine, f g components via EY and so there's a map F : W Ñ EY such that pF ✏ f g. Then g and F ensure a map g ✶ : W Ñ If such that vg ✶ ✏ g. ❭❬ subsequent we country with no facts the duals of numerous propositions of part three. 2. we start with precipitated maps of homotopy fibers. ninety two three Cofibrations and Fibrations Proposition three. three. 15 Given a homotopy-commutative diagram X  Y α / X✶ β  / Y✶ f✶ f such that f ✶ α ✔F βf. Then there exists a map ΨF : If following diagram i ΩY / If Ωβ  ΩY i✶  v /X v✶  / X, α ΨF / If Ñ If ✶ such that during the the left sq. is homotopy-commutative and the precise sq. is commutative, the place v and v ✶ are fiber maps and that i and that i✶ are inclusions.

6. 1 creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 2 Homotopy Pushouts and Pullbacks I . . . . . . . . . . . . . . . . . . . . . . 6. three Homotopy Pushouts and Pullbacks II . . . . . . . . . . . . . . . . . . . . . 6. four Theorems of Serre, Hurewicz, and Blakers–Massey . . . . . . . . . 6. five Eckmann–Hilton Duality II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . workouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 195 196 207 214 225 227 7 Homotopy and Homology Decompositions . . . . . . . . . . . . . . . . 7. 1 creation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Four. 2 The Coexact and particular series of a Map . . . . . . . . . . . . . . . . four. three activities and Coactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four. four Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred fifteen one hundred fifteen 116 a hundred twenty five a hundred thirty xi xii Contents four. five Homotopy teams II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred thirty five workouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a hundred and fifty five functions of Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. 1 creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Download PDF sample

Rated 4.82 of 5 – based on 43 votes