Vipul: Created page with "==Statement== Suppose $X$ is a topological space. Then, knowledge of each of the (singular) homology groups $H_n(X;\mathbb{Z})$ up to isomorphism d..."

2011-07-29T16:54:31Z

Created page with "==Statement== Suppose <math>X</math> is a topological space. Then, knowledge of each of the (singular) homology groups <math>H_n(X;\mathbb{Z})</math> up to isomorphism d..."

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==Statement==

Suppose <math>X</math> is a [[topological space]]. Then, knowledge of each of the (singular) [[homology group]]s <math>H_n(X;\mathbb{Z})</math> up to isomorphism determines each of the [[cohomology group]]s <math>H^n(X;\mathbb{Z})</math> up to isomorphism.

==Facts used==

# [[uses::Dual universal coefficients theorem]]

==Proof==

The proof follows directly from Fact (1), which provides an explicit computational procedure.

Homology groups determine cohomology groups - Revision history

Vipul: Created page with "==Statement== Suppose X is a topological space. Then, knowledge of each of the (singular) homology groups H_n(X;\mathbb{Z}) up to isomorphism d..."

Vipul: Created page with "==Statement== Suppose $X$ is a topological space. Then, knowledge of each of the (singular) homology groups $H_n(X;\mathbb{Z})$ up to isomorphism d..."