Download Algebraic Topology Poznań 1989: Proceedings of a Conference by C. Allday, V. Puppe (auth.), Stefan Jackowski, Bob Oliver, PDF

By C. Allday, V. Puppe (auth.), Stefan Jackowski, Bob Oliver, Krzystof Pawałowski (eds.)

ISBN-10: 3540540989

ISBN-13: 9783540540984

As a part of the clinical task in reference to the seventieth birthday of the Adam Mickiewicz college in Poznan, a global convention on algebraic topology was once held. within the ensuing court cases quantity, the emphasis is on vast survey papers, a few provided on the convention, a few written subsequently.

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Extra info for Algebraic Topology Poznań 1989: Proceedings of a Conference held in Poznań, Poland, June 22–27, 1989

Example text

This is the only book devoted entirely to material related to Hopf inv~riants. We can only describe it very briefly. The Hilton-Milnor process takes place in a free group, and in the collection process commutators ard produced. The book discusses commutator calculus in general. Then it derives results for distfibutivity laws and homotopy operations on spheres, including the higher Hopf invariaats. Then it uses the previous theory with different coefficient rings to deduce results about homotopy groups.

Illman has also announced [27] that the restriction homomorphism satisfies the natural asaociativity property rest" res. = res 4. Some geometric applications The product formula of the preceding section gives rise immediately to the following nice geometric result. _! Let G be a finite group and X a G-complex 6-domlnated by a finite one. Let V be any unitary complex representation of the group G and s(V) its unit sphere. Then the product X x s(V) with the diagonal Gaction has the G-homotopy type o f a finite G-complex.

Now following Liick [33], [34] one can introduce the notion of a universal functorial additive invariant (UFAI). Let C be a small full subeategory of the category of G-spaces containing 0 and {pt} . We assume that C is closed under G-homotopy equivalences and G-push-outs. A fuuctorial additive invaciant (FAI) (B,b) for C consists of a functor B : C --~ Ab from C to 27 the category of abelian groups and an assignement b associating to an object X in C an element b(X) E B(X) such that the following conditions are fullfilled: (a) If f : X --~ Y is a G-homotopy equivalence in C then B(f)(b(X)) = b(Y).

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