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One Remark on the Realizability of Singular Cohomology Groups

✍ Scribed by Dinh The Luc

Publisher: John Wiley and Sons
Year: 1978
Tongue: English
Weight: 87 KB
Volume: 82
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19780820108

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✦ Synopsis

But Hn(X; Z)xQ and E x t (Q, 2 ) is isomorphic with countable product of groups Q which implies Ext (Q, Z)xQNa (where K O is the smallest ii1finit.e cardinal number, see [2], IX), and therefore

(2) Substituting (2) into ( l ) , we obtain:

H"(X; 2) z E x t (Hom (H"-'(X; Z), 2 ) +QNo, 2) Z E x t (Hom (Hn-'(X; Z), Z), 2) + E x t (QNo, 2).

88 DINE T m Luo, One Remarks on the Realiztlbility of Singular Cohomology Groups

The sequence 0-@ Q-nQ-n&/@ Q -0 (where @ denotes the direct sum and n the direct product) gives the exact sequence in the application of the functor Ext (. , 2) :

No

Thus, Ext (nQ, 2) is not countable, and H n ( X ; Z), being direct sum (3), is not countable. This contradicts Bn(X; 2) x& and thus our proposition is proved.

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