reduced cylinder in nLab

Contents

Idea

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

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

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

X∧(I +)∈Top */. X \wedge (I_+) \;\; \in Top^{\ast/} \,.

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

X∧(I +)≃(X×I)/({x}×I). 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}×I\{x\} \times I is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as XX is a compact Hausdorff space.

References

Early lecture notes include

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