The topological mapping cone of is , where is the cone on and its base is identified with through . For nonempty , the cone vertex supplies a distinguished basepoint. This construction is related to, but distinct from, the mapping cone of a chain map.
For a cellular map between nonempty CW complexes, the reduced cellular chain complex of its topological mapping cone has differentialHere and the degree-zero group identifies a vertex with the reduced chain , where is the cone vertex. Thus this is the mapping cone of , in the displayed ordering of the summands.
For nonempty , the topological mapping cone has this long exact sequence in homology, with ordinary homology for and reduced homology for . In degree zero, the augmentation of is accounted for by the cone vertex. The sequence follows from the Mayer–Vietoris theorem applied to neighborhoods of the cone and .
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In algebraic topology, a mapping cone is a construction associated with a continuous map between two topological spaces. It is often used in the context of homology and cohomology theories, especially in the study of fiber sequences, and it is significant in understanding the relationships between different topological spaces.