# Analytic geometry and principles of algebra by Alexander Ziwet, Louis Allen Hopkins

By Alexander Ziwet, Louis Allen Hopkins

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**Example text**

2. k E3 Then ((zO,zl) TnOk O. ei, ~2) 1 E yo S3 f ( zo'z ) = (pe 2 To show that the with 0, is a hemisphere and thus a closed disk. (0,1) 1 3 0 S3 G is cellular and satisfies obvious map p ~ as Euclidean 4-space, by constructing a condition (-K-). Let =0 p3 1(p,q) for which xo is a closed 2-disk, and E3 lie COmpute the homology of yo 2 C and regard Aq 1, then is xo + iyo then ~ P Zo form a cyclic group s3/ G In the special case = S3 If we set m = 0 mod p • Hence l-1oreover G is a group ~(~ok) G satisfies condi above.

DEFINITION. 12 and Im 0 pectively. The qth cohomology module of q Zq/B and is denoted Hq(c). The collection of cohomology modules of ((Cq},O) C Let Z. G, written 65. o'm q"'q for each q properties (1) - (3) for CHAPTER III clude that i f REGUlAR COMPLEXES WITH IDENTIFICATIONS f: Accordingly, if 1. That is, if of equal dimension. 1. a0 carries o From properties (3) and (4) con we a ~:r and g::r ~ a are in F then g= f- l f: a ~:r and h: a ~:r are in F then h = f. The identifications Let K be a regular complex, and let of K F f and of a a < a o K if whenever ~ onto a (closed) face of ~ be cells onto a onto is ~ a "-' ~.

1. to be the collection of maps obtained by restricting 68. fA. 7) to the closed cells of has one cell in each dimension. K. a face of The The incidence :t'unction given on pages 36 and 37 of Chapter I I is invariant under F'. LEMMA. 4. complex K. Let F Then th~re e~ists is invariant under be a family of identifications for a regular an incidence fUnction on K that K. n a F. 6. As 0 al 1, choose a l-cell from each equivalence class Call the vertices of a Ao and Bo For each T a is invariant on cells of and 1 --;..