Talk:Mapping cone (topology)

K-Theory

Something about the mapping cone of a map of c*-algebras is missing. cf. N.E. Wegge-Olsen: K-Theory and C*-Algebras.

Beyond ridiculous

This is the first sentence of the article:

""In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space."

It is beyond ridiculous to call the mapping cone "analogous to" a quotient space, because a mapping cone is a quotient space.

Coarse intuition removed

I removed the following paragraph for now:

Coarsely, one is taking the quotient space by the image of X, so $C_{f}=Y/f(X)$; this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected map.

I understand the sentiment, but this intuition ignores the crucial importance of the map f; it seems to imply (as does the image accompanying our article) that only the image of f matters. Consider for instance the map f wrapping the closed unit interval I=[0,1] around the unit circle S¹, f(t)=exp(2πit). In this case, f(I) is not collapsed to a point, but instead C_f is homotopy equivalent to S¹. If instead we consider the identity g: S¹ → S¹, then C_f is homotopy equivalent to a point. The images of f and g are identical. AxelBoldt (talk) 14:48, 28 December 2024 (UTC)