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Cellular homology can also be used to calculate the homology of the genus g surface.The fundamental polygon of is a -gon which gives a CW-structure with one 2-cell, 1-cells, and one 0-cell.
Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a homology theory. To compute an extraordinary (co)homology theory for a CW complex, the Atiyah–Hirzebruch spectral sequence is the analogue of cellular homology. Some examples:
In biology, homology is similarity in anatomical structures or genes between organisms of different taxa due to shared ancestry, regardless of current functional differences. Evolutionary biology explains homologous structures as retained heredity from a common ancestor after having been subjected to adaptive modifications for different ...
3. The cellular homology is the (canonical) homology of a CW complex. Note it applies to CW complexes and not to spaces in general. A cellular homology is highly computable; it is especially useful for spaces with natural cell decompositions like projective spaces or Grassmannian. chain homotopy
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,
Homology-directed repair (HDR) is a mechanism in cells to repair double-strand DNA lesions. [1] The most common form of HDR is homologous recombination . The HDR mechanism can only be used by the cell when there is a homologous piece of DNA present in the nucleus , mostly in G2 and S phase of the cell cycle .
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In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant , and is in fact isomorphic to singular homology .