reduced cylinder

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.

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.

Early lecture notes include

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

Last revised on June 18, 2017 at 05:07:26. See the history of this page for a list of all contributions to it.