By Tammo Tom Dieck
This ebook is written as a textbook on algebraic topology. the 1st half covers the cloth for 2 introductory classes approximately homotopy and homology. the second one half offers extra complex purposes and ideas (duality, attribute periods, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy concept. For this function, classical effects are offered with new simple proofs. on the other hand, you can commence extra normally with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, phone complexes and fibre bundles. a distinct function is the wealthy offer of approximately 500 workouts and difficulties. numerous sections comprise themes that have now not seemed prior to in textbooks in addition to simplified proofs for a few vital effects. necessities are general aspect set topology (as recalled within the first chapter), straight forward algebraic notions (modules, tensor product), and a few terminology from type thought. the purpose of the booklet is to introduce complex undergraduate and graduate (master's) scholars to simple instruments, options and result of algebraic topology. enough heritage fabric from geometry and algebra is incorporated. A book of the ecu Mathematical Society (EMS). disbursed in the Americas by way of the yank Mathematical Society.
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Additional resources for Algebraic Topology (EMS Textbooks in Mathematics)
The importance of this notion is seen from several facts. (1) The classical tools of algebraic topology are functors from a category of spaces to an algebraic category, say of abelian groups. , homotopic maps have the same value under the functor. (2) One can change maps by homotopies and spaces by homotopy equivalences. This fact allows for a great flexibility. But still global geometric information is retained. Basic principles of topology are deformation and approximation. One idea of deformation is made precise by the notion of homotopy.
B. A morphism from p W X ! B to q W Y ! B is a map f W X ! Y such that qf D p. If B is a point, then TOPB can be identified with TOP, since each space has a unique map to a point. If K D f g is a point, then TOPK is the same as TOP0 . If p W X ! b/ is called the fibre of p over b; in this context, B is the base space and X the total space of p. A map in TOPB will also be called fibrewise or fibre preserving. Categories like TOPK or TOPB have an associated notion of homotopy. A homotopy H t is in TOPK if each H t is a morphism in this category.
A pullback of a shrinkable map is shrinkable. A pushout of a deformation retract is a deformation retract. 14. Let Y; Z be compact or X; Z be locally compact. X ^ Y / ^ Z ! Y ^ Z/ is a homeomorphism. ) the map is always a homeomorphism. ) 15. Aj ^ B/ ! canonical map which is on each sumj Aj ^ B be theW mand Ak ^ B induced by the inclusion Ak ! j Aj . Show that this map is a homeomorphism if the index set is finite. Aj B/. 16. Let A be a compact subset of X and p W X ! X=A be the quotient map. Y / is a quotient map.
Algebraic Topology (EMS Textbooks in Mathematics) by Tammo Tom Dieck