By Tom Dieck T.

This ebook 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 provides extra complex functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy concept. For this objective, classical effects are provided with new ordinary proofs. however, you can still commence extra normally with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, mobilephone complexes and fibre bundles. a different characteristic 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 typical element set topology (as recalled within the first chapter), straight forward algebraic notions (modules, tensor product), and a few terminology from classification idea. the purpose of the e-book is to introduce complicated undergraduate and graduate (master's) scholars to simple instruments, ideas and result of algebraic topology. adequate historical past fabric from geometry and algebra is integrated. A booklet of the ecu Mathematical Society (EMS). dispensed in the Americas through the yank Mathematical Society.

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Then l W G G=H ! x; gH / 7! xgH is a continuous action. 2/ G=H is separated if and only if H is closed in G. In particular, G is separated if feg is closed. 3/ Let H be normal in G. Then the factor group G=H with quotient topology is a topological group. A space G=H with the G-action by left multiplication is called a homogeneous space. The space of left cosets Hg is H nG; it carries a right action. 3) Example. Homogeneous spaces are important spaces in geometry. A; v/ 7! Av. The action is transitive.

S n ! x; y/ 2 S n S n j x 6D yg, x 7! x; x/ is an h-equivalence. 6. Let f; g W X ! x/. Then f ' g. 7. Let A E n be star-shaped with respect to 0. Show that S n 1 Rn XA is a deformation retract. 8. x; v/ 2 S n RnC1 j x ? vg ! x; v/ 7! x is called the tangent bundle of S n . Show that p admits a fibrewise homeomorphism with pr W S n S n X D ! x; y/ 7! x (with D the diagonal). 4 Mapping Spaces and Homotopy It is customary to endow sets of continuous maps with a topology. In this section we review from point-set topology the compact-open topology.

X; gx/ j x 2 X; g 2 Gg is an equivalence relation on X . The set of equivalence classes X mod R is denoted by X=G. The quotient map q W X ! X=G is used to provide X=G with the quotient topology. The resulting space X=G is called the orbit space of the G-space X . A more systematic notation for the orbit space of a left action would be GnX . The equivalence class of x 2 X is the orbit Gx through x. An action is transitive if it consists of a single orbit. The set Gx D fg 2 G j gx D xg is a subgroup of G, the isotropy group or the stabilizer of the G-space X at x.