nLab reduced cylinder

Idea

The concept of reduced cylinder is the analog for pointed objects of cylinder constructions.

The mapping cone of $X \to \ast$ formed with the standard reduced cylinder is the reduced suspension of $X$.

Applying the reduced cylinder construction degreewise to a sequential spectrum yields the standard cylinder spectrum construction.

For pointed topological spaces

Specifically in topological spaces, with $I = [0,1] \subset \mathbb{R} \in$ Top the closed interval with its Euclidean metric topology and $I_+ \in Top^{\ast/}$ its pointed version with a basepoint freely adjoined, then for $X$ a pointed topological space, the standard reduced cylinder over it is the smash product

$$X \wedge (I_+) \;\; \in Top^{\ast/} \,.$$

This is obtained from the ordinary standard cylinder $X \times I$ by passing to the quotient space (this example) given by collapsing the copy of $I$ that sits over the basepoint $x$ of $X$:

$$X \wedge (I_+) \simeq (X \times I)/(\{x\} \times I) \,.$$

For the purposes of generalized (Eilenberg-Steenrod) cohomology theory typically it does not matter whether one evaluates on the standard cylinder or the reduced cylinder. For example for topological K-theory since since $\{x\} \times I$ is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as $X$ is a compact Hausdorff space.

References

Early lecture notes include

• Frank Adams, part III, section 2 Stable homotopy and generalised homology, 1974