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To take the homology of a chain complex, one starts with a chain complex, which is a sequence (,) of abelian groups (whose elements are called chains) and group homomorphisms (called boundary maps) such that the composition of any two consecutive maps is zero:
By Wu's formula, a spin 4-manifold must have even intersection form, i.e., (,) is even for every x. For a simply-connected smooth 4-manifold (or more generally one with no 2-torsion residing in the first homology), the converse holds. The signature of the intersection form is an important invariant.
Cellular homology can also be used to calculate the homology of the genus g surface. The fundamental polygon of Σ g {\displaystyle \Sigma _{g}} is a 4 n {\displaystyle 4n} -gon which gives Σ g {\displaystyle \Sigma _{g}} a CW-structure with one 2-cell, 2 n {\displaystyle 2n} 1-cells, and one 0-cell.
Synapomorphy/homology – a derived trait that is found in some or all terminal groups of a clade, and inherited from a common ancestor, for which it was an autapomorphy (i.e., not present in its immediate ancestor). Underlying synapomorphy – a synapomorphy that has been lost again in many members of the clade. If lost in all but one, it can ...
For all integers r ≥ r 0, an object E r, called a sheet (as in a sheet of paper), or sometimes a page or a term, Endomorphisms d r : E r → E r satisfying d r o d r = 0, called boundary maps or differentials, Isomorphisms of E r+1 with H(E r), the homology of E r with respect to d r. The E 2 sheet of a cohomological spectral sequence
Synapomorphy/Homology – a derived trait that is found in some or all terminal groups of a clade, and inherited from a common ancestor, for which it was an autapomorphy (i.e., not present in its immediate ancestor). Underlying synapomorphy – a synapomorphy that has been lost again in many members of the clade. If lost in all but one, it can ...
In general one uses singular homology; but if X and Y happen to be CW complexes, then this can be replaced by cellular homology, because that is isomorphic to singular homology. The simplest case is when the coefficient ring for homology is a field F. In this situation, the Künneth theorem (for singular homology) states that for any integer k,
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1 ...