Elements of Homotopy TheorySpringer Science & Business Media, 2012 M12 6 - 746 páginas As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious. |
Contenido
1 | |
3 | |
9 | |
13 | |
17 | |
21 | |
Filtered Spaces | 27 |
Fibrations 1 3 | 30 |
Fibrations Having a Sphere as Fibre | 349 |
The Homology Sequence of a Fibration | 363 |
Proof of the Suspension Theorem | 371 |
The mod 2 Steenrod Algebra | 394 |
Some Relations among the Steenrod Squares | 403 |
Chapter IX | 415 |
The Postnikov Invariants of a Space | 421 |
Reconstruction of a Space from its Postnikov System | 430 |
13 | 31 |
17 | 35 |
21 | 37 |
CWcomplexes | 46 |
Construction of CWcomplexes 2 Homology Theory of CWcomplexes 3 Compression Theorems 4 Cellular Maps 5 Local Calculations 6 Regular Cell ... | 47 |
Appendix | 59 |
Products and the Cohomology Ring | 88 |
Chapter III | 96 |
Homotopy and the Fundamental Group | 98 |
Groups of Homotopy Classes | 115 |
Hspaces | 116 |
Hspaces | 121 |
Exact Sequences of Mapping Functors | 127 |
Homology Properties of Hspaces and Hspaces | 142 |
Hopf Algebras | 149 |
Chapter IV | 157 |
Relative Homotopy Groups | 158 |
The Homotopy Sequence | 161 |
The Operations of the Fundamental Group on the Homotopy Sequence | 164 |
The Hurewicz Map | 166 |
The Eilenberg and Blakers Homology Groups | 170 |
The Homotopy Addition Theorem | 174 |
The Hurewicz Theorems | 178 |
Spaces with Base Points 98 | 181 |
Homotopy Relations in Fibre Spaces | 185 |
Fibrations in Which the Base or Fibre is a Sphere | 194 |
Elementary Homotopy Theory of Lie Groups and Their Coset Spaces | 196 |
Chapter V | 209 |
The Effect on the Homotopy Groups of a Cellular Extension | 211 |
Spaces with Prescribed Homotopy Groups | 216 |
Weak Homotopy Equivalence and CWapproximation | 219 |
Aspherical Spaces | 224 |
Obstruction Theory | 228 |
Homotopy Extension and Classification Theorems | 235 |
EilenbergMac Lane Spaces | 244 |
Cohomology Operations | 250 |
Chapter VI | 255 |
Bundles of Groups | 257 |
Homology with Local Coefficients | 265 |
Computations and Examples | 275 |
Local Coefficients in CWcomplexes | 281 |
Obstruction Theory in Fibre Spaces | 291 |
The Primary Obstruction to a Lifting | 297 |
Characteristic Classes of Vector Bundles | 305 |
Elementary Theory | 314 |
The James Reduced Products | 326 |
Further Properties of the Wang Sequence | 336 |
Relative Postnikov Systems | 443 |
Postnikov Systems and Obstruction Theory | 449 |
Chapter X | 456 |
Nilpotency of X | 465 |
Chapter XI | 479 |
Homotopy Operations | 488 |
Homotopy Operations | 490 |
The Hopf Invariant | 494 |
The Functional Cup Product | 496 |
The Hopf Construction | 502 |
Geometrical Interpretation of the Hopf Invariant | 507 |
The HiltonMilnor Theorem | 511 |
Proof of the HiltonMilnor Theorem | 515 |
The HopfHilton Invariants | 533 |
Chapter XII | 542 |
The Whitehead Product | 543 |
Homotopy Properties of the James Imbedding | 544 |
Suspension and Whitehead Products | 546 |
The Suspension Category | 550 |
Group Extensions and Homology | 561 |
Stable Homotopy as a Homology Theory | 571 |
Comparison with the EilenbergSteenrod Axioms | 578 |
Cohomology Theories | 594 |
Chapter XIII | 602 |
The Homology of a Filtered Space | 604 |
Exact Couples | 609 |
The Exact Couples of a Filtered Space | 613 |
The Spectral Sequence of a Fibration | 623 |
Proofs of Theorems 4 7 and 4 8 | 632 |
The AtiyahHirzebruch Spectral Sequence | 640 |
The LeraySerre Spectral Sequence | 645 |
Multiplicative Properties of the LeraySerre Spectral Sequence | 654 |
Further Applications of the LeraySerre Spectral Sequence | 668 |
Appendix A Compact Lie Groups | 673 |
Subgroups Coset Spaces Maximal Tori | 674 |
Classifying Spaces | 678 |
The Spinor Groups | 680 |
The Cayley Algebra K | 686 |
Automorphisms of K | 690 |
The Exceptional Jordan Algebra J | 695 |
The Exceptional Lie Group F4 | 701 |
Additive Relations | 716 |
730 | |
102 | 735 |
737 | |
741 | |
Otras ediciones - Ver todas
Términos y frases comunes
0-connected 1)-connected A₁ abelian group algebra An+1 b₁ B₂ base point boundary operator cell chain complex Chapter characteristic map cochain coefficients cohomology operation commutative diagram compactly composite Corollary CW-complex defined direct sum E₁ element epimorphism exact sequence extension f₁ fibration fibre F finite follows functor ƒ₁ g₁ H-space h₁ H₁(X h₂ H₂(X Hence homology groups homology theory homomorphism homotopy class homotopy lifting homotopy sequence homotopy type Hopf Hq(X Hurewicz Theorem i₁ identity map inclusion map induced injection integer isomorphism j₁ K(II k₁ Lemma Let f Let G Let h map f map ƒ map g map h monomorphism Moreover morphism n-cell NDR-pair P₁ pair partial lifting phism proof properties prove relative CW-complex represents simplicial space subcomplex subgroup subset subspace Suppose Theorem Let topology weak homotopy equivalence Whitehead X₁ Xn+1 Z₂