By Richard M. Kane
This exposition of the idea of finite Hopf areas info the advance of the topic during the last thirty years, with the homology of such areas as its major subject matter. the 3 leader parts of research within the quantity are: - The examine of finite H-spaces with torsion unfastened vital homology. - The examine of finite H-spaces with homology torsion. - the development of finite H-spaces.
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Bx Ex §6-3: Projective Planes The preceeding study of loop spaces and classifying spaces can be car- 52 The Homology of Hopf Spaces ried further. One can break down the previous classifying spaces into a series of "projective planes" The definition of Pn(X) are as follows (topological groups) Pn(X) =G* (associative H-space) Pn(X) = U As G *... * s x X/ s~O (Am spaces) Pn(X) =U K GIG ;:: s s~O s+ 2 x XI ;:: The term "projective plane" arises from the fact that. S1 and S3. we have the identities P (SO) n p (S1) n p (S3) n = IRpn = lCpn = Hpn The theory of projective planes blends nicely with Stasheff's theory of An structures.
If the group operations are cellular then E and B are CW comG G plexes. Observe that we must then have flBG '" G For B CW implies that flB is CW as well. Also. the long exact sequence in G G IT*( ) plus the fact that E is contractible implies IT*(flBG) ~ IT*(G). G Remark: As with Lie groups the classifying space B acquires its name from G the fact that it classifies principal G bundles over finite X. As with Lie groups such bundles are in 1-1 correspondence wi th the homotopy classes (i) Join construction To construct contractible fibre bundles we require the join construction.
More p precisly, let~: YAY ~ SN be the S-dual map. (Y;W ) ~ 1 for all i. It follows that for all i. Such a symmetry holds for ~* H (X+ ;W ) p P = H* (X;Wp ). ::'::N-i H' ~ (Y;W) p This follows from the arguments in §4-2. For, if X is a Poincare complex of dimension n then the non singular form ( , ): Hi(X;W ) 0 Hn-i(X;W ) ~ W p p p defined in §4-2 establishes the isomorphisms On the other hand, i f we work wi th X rather . than X+ then this symmetry = 0 whIle H (X;Wp) = Wp' p) The above discussion also gives a very strong hint regarding how to ~O would be destroyed since,now, H (X;W prove the theorem.