Algebraic Topology (EMS Textbooks in Mathematics) by Tammo Tom Dieck

By Tammo Tom Dieck

This booklet is written as a textbook on algebraic topology. the 1st half covers the fabric for 2 introductory classes approximately homotopy and homology. the second one half offers extra complicated functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory path with homotopy thought. For this objective, classical effects are awarded with new uncomplicated proofs. however, you could commence extra often with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, mobilephone complexes and fibre bundles. a distinct characteristic is the wealthy provide of approximately 500 routines and difficulties. numerous sections comprise themes that have no longer seemed sooner than in textbooks in addition to simplified proofs for a few vital effects. necessities are commonplace aspect set topology (as recalled within the first chapter), user-friendly algebraic notions (modules, tensor product), and a few terminology from type thought. the purpose of the e-book is to introduce complicated undergraduate and graduate (master's) scholars to uncomplicated instruments, techniques and result of algebraic topology. adequate history fabric from geometry and algebra is incorporated. A booklet of the ecu Mathematical Society (EMS). disbursed in the Americas via the yankee Mathematical Society.

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Extra resources for Algebraic Topology (EMS Textbooks in Mathematics)

Example text

The choice of a base point is an additional structure. , to X with an additional base point added (topological sum), with the obvious extension to pointed maps. This functor is compatible with homotopies. We also have the forgetful functor ˇ from TOP0 to TOP. X; ˇY /, and similarly for the homotopy categories. The category TOP0 has sums and products. Xj ; xj / is a family of pointed spaces. xj / of base points Wis taken as base point in the`product Q j Xj ; this yields the pointed product. Let j 2J Xj be the quotient of j 2J Xj where all base points are W identified to a single new base point.

Z; X/ ! Y; Z/ ! X; Z/ are h-equivalences. Z; X / ! Y; Z/ ! X; Z/ are pointed h-equivalences. Problems 1. Verify that f Z and Z f are continuous. 2. An inclusion i W Z Y`induces an embedding i X W ZX ! Y X . Q 3. Xj ; Y / is always a homeomorphism. Q 4. X; j Yj / ! X; Yj /, f 7! prj f / is always bijective and continuous. If X is locally compact, it is a homeomorphism. 5. Let p W X ! Y be a surjective continuous map. Suppose the pre-image of a compact set is compact. Then Z p W Z Y ! Z X is an embedding.

CnC1 /. Finally, one can define the quaternionic projective space HP n in a similar manner as a quotient of HnC1 X 0 or as a quotient of S 4nC3 . We generalize projective spaces. Let W be an n-dimensional real vector space. W / the set of k-dimensional subspaces of W . W /. Suppose W carries an inner product. w1 ; : : : ; wk / in W considered as a subspace of W k . W / the Stiefel manifold of orthonormal k-frames in W . W / ! w1 ; : : : ; wk / to the subspace Œw1 ; : : : ; wk  spanned by this sequence.

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