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This incorporates the Abel Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian.
Notice here that the differential lowers the degree and so this differential graded Lie algebra is considered to be homologically graded.
where Z ^ k ( X ) denotes the group of algebraic cycles of some fixed codimension " k " and the subscripts indicate the groups that are homologically trivial, respectively algebraically equivalent to zero.
The choice of ( co ) homological grading usually depends upon personal preference or the situation as they are equivalent : a homologically graded space can be made into a cohomological one via setting L ^ i = L _ {-i }.
Similarly, the subgroup that acts trivially on the homology of " M " is called the "'Torelli group "'of " M ", one could think of this as the mapping class group of a homologically-marked surface.
It follows that a compact K�hler Einstein manifold " X " must have canonical bundle " K " " X " either anti-ample, homologically trivial, or Calabi Yau, or with ample canonical bundle ( which implies general type ), respectively.
In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the "'Adams filtration "'and the "'Adams-Novikov filtration "'allow a stable homotopy group to be understood as built from layers, the " n " th layer containing just those maps which require at most " n " auxiliary spaces in order to be a composition of homologically trivial maps.