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This implies that has trivial fundamental group, so as a consequence, it also has trivial first homology group. The torus T 2 {\displaystyle T^{2}} has closed curves which cannot be continuously deformed into each other, for example in the diagram none of the cycles a , b or c can be deformed into one another.
A special case of the Hurewicz theorem asserts that the first singular homology group is, colloquially speaking, the closest approximation to the fundamental group by means of an abelian group. In more detail, mapping the homotopy class of each loop to the homology class of the loop gives a group homomorphism
In terms of group homology, a perfect group is precisely one whose first homology group vanishes: H 1 (G, Z) = 0, as the first homology group of a group is exactly the abelianization of the group, and perfect means trivial abelianization. An advantage of this definition is that it admits strengthening:
If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H 2 (G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation).
A n is generated by 3-cycles – so the only non-trivial abelianization maps are A n → Z 3, since order-3 elements must map to order-3 elements – and for n ≥ 5 all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the ...
A homology G-manifold (without boundary) of dimension n over an abelian group G of coefficients is a locally compact topological space X with finite G-cohomological dimension such that for any x∈X, the homology groups (,,) are trivial unless p=n, in which case they are isomorphic to G.
Homology groups are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are often very complex and hard to compute. In contrast, homology groups are commutative (as are the higher homotopy groups). Hence, it is sometimes said that "homology is a commutative alternative to homotopy". [7]
Considering homology groups with other coefficients leads to other definitions of connectivity. For example, X is F 2-homologically 1-connected if its 1st homology group with coefficients from F 2 (the cyclic field of size 2) is trivial, i.e.: (;).