homology group

homology group

[hə′mäl·ə·jē ‚grüp]
(mathematics)
Associated to a topological space X, one of a sequence of Abelian groups Hn (X) that reflect how n-dimensional simplicial complexes can be used to fill up X and also help determine the presence of n-dimensional holes appearing in X. Also known as Betti group.
References in periodicals archive ?
(iv) As a result of [partial derivative] * [partial derivative] = 0, one can define the pth homology group [H.sub.p](K) as the quotient of the p-cycles [Z.sub.p], elements of [C.sub.p] which are mapped to 0 by [partial derivative], and p-boundaries [B.sub.p], which is the image of [C.sub.p+1] under [partial derivative].
The homology group [H.sub.*]([M.sub.K]; Z) = [H.sup.0]([M.sub.K]; Z) [[direct sum] [H.sub.1]([M.sub.K]; Z) has the basis {[p], [[mu]]}, where [p] is the homology class of a point and [[mu]] is that of the meridian of K.
The 13 selected peer-reviewed papers explore such aspects of logic as an analogy between cardinal characteristics and highness properties of oracles, a non-uniformly C-productive sequence and non-constructive disjunctions, the characterization of the second homology group of stationary type in a stable theory, some questions concerning ab initio generic structures, realizability and existence property of a constructive set theory with types, a goal-directed unbounded coalitional game and its complexity, large cardinals and higher degree theory, and degree spectra of equivalence relations.
It is known from  and  that in this case the homology group [H.sub.r](X, Y, Z) = 0 if r [greater than or equal to] n + q and, is torsion free if r = n + q - 1.
In this setting, the concept of size function coincides with the dimension of the 0-th multidimensional persistent homology group, i.e., the 0-th rank invariant (Carlsson and Zomorodian, 2007).
and transfer of seven species of the genus Pseudomonas homology group II to the new genus with type species Burkholderia cepacia (Palleroni & Holmes 1981) comb.
(2) Compute geometric realizations of a set of generators that form a basis of the first and second homology groups of K (for the zeroth homology group, this is trivial) in O([n.sup.2][bar]g) and O(n) time (and space), respectively, where [bar]g is an invariant of K such that [bar]g [is less than] n always.
* [H.sup.[kappa].sub.q] (X, A) = [Z.sup.[kappa].sub.q](X, A)/[B.sup.[kappa].sub.q] (X, A) is called the qth digital relative simplicial homology group.
Changing a basis of the first homology group [H.sub.1]([h.sub.1]) (resp.
Calderbank, Hanlon and Robinson  extended these results by considering the action of the symmetric group [G.sub.n-1] on the top homology group of the order complex of [[PI].sup.d.sub.n] - {[??]}.
heilmannii" type 2 were highly related and formed a distinct cluster within the rRNA homology group III (i.e., the Helicobacter phylogenetic branch) of rRNA super-family VI (data not shown).
In fact, it computes this homotopy group as a homology group of another space (simplicial set): [K.sub.4].

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