The concept of reduced cylinder is the analog for pointed objects of cylinder constructions.
The mapping cone of formed with the standard reduced cylinder is the reduced suspension of .
Applying the reduced cylinder construction degreewise to a sequential spectrum yields the standard cylinder spectrum construction.
Specifically in topological spaces, with Top the closed interval with its Euclidean metric topology and its pointed version with a basepoint freely adjoined, then for a pointed topological space, the standard reduced cylinder over it is the smash product
This is obtained from the ordinary standard cylinder by passing to the quotient space (this example) given by collapsing the copy of that sits over the basepoint of :
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 is a contractible closed subspace, then this prop. says that topological vector bundles do not see a difference as long as is a compact Hausdorff space.
Early lecture notes include
Last revised on June 18, 2017 at 09:07:26. See the history of this page for a list of all contributions to it.