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Torsion in Homology and Cohomology

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Describe the Universal Coefficient Theorem for homology.

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The Universal Coefficient Theorem for homology provides a relationship between the homology of a space and its cohomology. Specifically, it states that the nth cohomology group is isomorphic to the Hom group of the nth homology group with integers, plus the Ext group of the (n-1)th homology group with integers.

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Provide a statement of the Universal Coefficient Theorem for cohomology.

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The Universal Coefficient Theorem for cohomology states that there is a short exact sequence

0ightarrowExt(Hn1(X),extbfZ)ightarrowHn(X)ightarrowHom(Hn(X),extbfZ)ightarrow00 ightarrow Ext(H_{n-1}(X), extbf{Z}) ightarrow H^n(X) ightarrow Hom(H_n(X), extbf{Z}) ightarrow 0
which splits, but not naturally.

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Define what a torsion coefficient is in homology.

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A torsion coefficient of an element in a homology group is the smallest positive integer that, when multiplied with the said element, yields the zero element of the group.

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How is the torsion subgroup of a homology group defined?

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The torsion subgroup of a homology group is the subgroup consisting of all the torsion elements of that homology group. It includes all elements that have a finite order.

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What role does torsion play in the computation of homology groups?

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In the computation of homology groups, torsion elements play a critical role in characterizing the group structure beyond free components. The presence of torsion reflects the presence of 'holes' or 'twists' in the space that are of finite order.

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Explain 'torsion in cohomology' in your own words.

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Torsion in cohomology refers to torsion elements within cohomology groups, similar to those in homology groups. These are elements that when paired with a certain nonzero cochain and evaluated, gives a zero in cohomology.

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What is torsion in the context of homology groups?

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Torsion in homology refers to the elements of finite order in an abelian group. Specifically, in the homology group, these are non-zero elements that, when multiplied by some non-zero integer, give the zero element.

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What is the significance of torsion in topological spaces?

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Torsion in topological spaces can reveal important topological features such as whether the space is simply-connected or not. It also helps in understanding the space's structure and classifying spaces up to homotopy equivalence.

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