By W.N. Bailey

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Z , •.. z ) . o n For an alternative description of lens spaces, and for =re information, the reader should consult Hilton and 1'l ylie's Ho=logy The space denoted by Theory, page 223. cpl 8. S2n+ljsl Cpn. is called complex pro,jective n-space, and is The space is a 2-sphere. •. ,zk' O, •.. ,O)} of state a result which allows us to compute the ho=logy groups of and the inclusion certain spaces very quickly. of 81 . S2k+l n+l C . C s2n+l Ck + l Then 82. X 2k+l k+l S = C 2n+l nS is consistent with the action Thus we have a diagram of inclusions and projections: In Chapter VII, we define ho=logy groups f'or arbitrary spaces in such a way that whenever a space with the subspace carries a regular 83 .

E , An+l l 2 2 n < u+ 1 larges t integeT of a regular compl ex is a subcomplex. then 3. 6. COROLLARY. 7. OOROLIoARY. Ho (K) IKI RO(K) of Z I{here k no. of components of K. That is , if is homeomorphic to K ILl, ,t hen and L are complexes such Ho(K) ~ HO(L) . rhenever K DEFllUTIOH. A has a finite or Ide order them in a silIIple sequence union of L _ n l {Ln} inductively by Let j be the If j < n+ 1 , A. 1. contains contains T B. to The construction of a maximal t ree can be carried sequence of trees 4.

1 appears in Chapter IX. Let H2 "" 0 X. may in certain cases be computed from an irregular complex on is given by H But the ho=logy groups need not be computed from a regular complex on n C H3 "" Z . 2~1 denote complex n-space, and let sented as the unit sphere in HO "" Z , X agree with the Then for each q 2 Hq(X) q morphic to the free abelian group on the q-cells of' K ( p-l mi 1) 1 s# ~i=O ~ a = pso o of q-cell is contained in ° 2 sia - Ta ) = a2 defined for q cellular homology groups of t he complex.