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Let C denote a chain complex, a portion of which is shown below: ...->C_(n+1)->C_n->C_(n-1)->.... Let H_n(C)=kerpartial_n/Impartial_(n+1) denotes the nth homology group. Then ...

In a chain complex of modules ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->..., the module B_i of i-boundaries is the image of d_(i+1). It is a submodule of C_i and is ...

A homology class in a singular homology theory is represented by a finite linear combination of geometric subobjects with zero boundary. Such a linear combination is ...

In a chain complex of modules ...->C_(i+1)->^(d_(i+1))C_i->^(d_i)C_(i-1)->... the module Z_i of i-cycles is the kernel of d_i, which is a submodule of C_i.

The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, ...

A term used in category theory to mean a general morphism. The term derives from the Greek omicronmuomicron (omo) "alike" and muomicronrhophiomegasigmaiotasigma (morphosis), ...

One of the Eilenberg-Steenrod axioms which states that, if f:(X,A)->(Y,B) is homotopic to g:(X,A)->(Y,B), then their induced maps f_*:H_n(X,A)->H_n(Y,B) and ...

An n-dimensional manifold M is said to be a homotopy sphere, if it is homotopy equivalent to the n-sphere S^n. Thus no homotopy group can distinguish between M and S^n. The ...

The branch of algebraic topology which deals with homotopy groups. Homotopy methods can be used to solve systems of polynomials by embedding the polynomials in a family of ...

The 6-polyiamonds illustrated above.