By Fox R.H. (ed.), Spencer D.C. (ed.), Tucker A.W. (ed.)

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In any case he set about an investigation of the spaces. Just when homology theory of finite-dimensional separable and where he succeeded in extending to complexes the validity of the fixed-point formula is difficult to determine. In 1930 he over-optimistically claimed its validity for a compact metric dimension having finite Betti numbers [46,48]. But certainly by 1933, he had devised a simple and elegant proof for a complex which runs as follows [57]. into itself which has no fixed point. Let/ be a map of a complex space of A finite K K is taken so fine that each closed star simplicial triangulation of fails to intersect its image under /.

3). Topological in variance is proved readily, since chain maps f:K->K l9 g:K'->K[ determine a unique chain map H+(K is fxg:KxK'-+K l xK( cochain is maps leading . to f formula for the induced the K x VJ = (/%) x . (/x a) (gvj. Associativity of the external cross-product (for three complexes) entirely obvious. The commutation law is easily derived. Let TiKxK'^K'xKbe defined by T(z, x = (x f ) onto cells; hence it f , x). induces a unique chain map. Then A T maps cells simple induction based on dT = Td yields where p, q are the dimensions of cr, T.

The commutation law is easily derived. Let TiKxK'^K'xKbe defined by T(z, x = (x f ) onto cells; hence it f , x). induces a unique chain map. Then A T maps cells simple induction based on dT = Td yields where p, q are the dimensions of cr, T. This gives immediately T(u xv) = (-l)wvxu. 4) Thus the external cross-products are easily and uniquely defined, and all properties are readily derivable. Now comes the essential feature of Lefschetz's method: the derivation of the cohomology cup- K K K ->K x be a complex, let d product from the cross-product.