The disc-to-cylinder map of a cell reattachment #
Comparing two attachments of a cell along homotopic attaching maps uses a map from the disc
D^k to the cylinder I × D^k (Milnor, Morse Theory, Lemma 3.6). Its time coordinate
TauCeti.TopCat.reattachmentTime is 1 on the disc of radius 1/2 and 2 - 2‖x‖ on the annulus
1/2 ≤ ‖x‖, and its radial coordinate TauCeti.TopCat.reattachmentRadial is 2x on the disc of
radius 1/2 and x / ‖x‖ on the annulus. Reading a homotopy K : I × D^k → Y of characteristic
maps along x ↦ (reattachmentTime x, reattachmentRadial x) gives the composite of a reattachment
map with the comparison map of the target attachment.
The straight-line homotopy TauCeti.TopCat.reattachmentPath in the convex cylinder joins this map
to x ↦ (0, x) and fixes the boundary sphere, where both maps are x ↦ (0, x). Read through K,
it is the homotopy relative to the attached base from that composite to the comparison map of the
source attachment.
References #
- J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Lemma 3.6.
The radial part x ↦ x / max(1/2, ‖x‖) of the disc-to-cylinder map of a reattachment: it is
2x on the disc of radius 1/2 and x / ‖x‖ outside it. It is the end of the radial push
TauCeti.TopCat.diskRadialPush with radius 1/2.
Instances For
On the annulus 1/2 ≤ ‖x‖, the radial part of x is its direction x / ‖x‖ on the boundary
sphere.
The point (1/2) • θ of the disc, for θ on the boundary sphere, has radial part θ.
The time part x ↦ 2 - 2‖x‖, clamped to [0, 1], of the disc-to-cylinder map of a
reattachment: the reversed radial parameter of the annulus 1/2 ≤ ‖x‖. It is 1 on the disc of
radius 1/2 and 0 on the boundary sphere.
Instances For
The straight-line homotopy in the cylinder I × D^k from
x ↦ (reattachmentTime x, reattachmentRadial x) at time 0 to x ↦ (0, x) at time 1. Its disc
coordinate is the radial push TauCeti.TopCat.diskRadialPush with radius 1/2, run backwards.