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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 ...
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:
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
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,
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.
This is different from homology, which is the term used to characterize the similarity of features that can be parsimoniously explained by common ancestry. [1] Homoplasy can arise from both similar selection pressures acting on adapting species, and the effects of genetic drift .
Hennig coined the key terms synapomorphy, symplesiomorphy, and paraphyly. He also asserted, in his "auxiliary principle", that "the presence of apomorphous characters in different species 'is always reason for suspecting kinship [i.e., that species belong to a monophyletic group], and that their origin by convergence should not be presumed a ...