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A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
Also, every integral cohomology class on a manifold can be represented by a "pseudomanifold", that is, a simplicial complex that is a manifold outside a closed subset of codimension at least 2. For a smooth manifold X , de Rham's theorem says that the singular cohomology of X with real coefficients is isomorphic to the de Rham cohomology of X ...
The singular homology groups H n (X) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex. Cohomology groups are formally similar to homology groups: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted , point in the direction of ...
An example is the chain complex defining the simplicial homology of a finite simplicial complex. ... The cohomology of this complex is called the de Rham cohomology of M.
For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the ...
The cohomology groups of X are defined as the homology groups of this complex; in a quip, "cohomology is the homology of the co [the dual complex]". The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows:
The Mayer–Vietoris sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for those theories satisfying the Eilenberg–Steenrod axioms, and it has variations for both reduced and relative (co)homology. Because the (co)homology of most spaces ...
The cohomology functors of ordinary cohomology theories are represented by Eilenberg–MacLane spaces. On simplicial complexes, these theories coincide with singular homology and cohomology. Homology and cohomology with integer coefficients.