I need to deformation retract a thickened Klein bottle to a 2D polyhedron.
I'm visualising the thickened Klein bottle as in the attached image - but I'm having trouble visualising the retraction. Can someone help?
According to your image, it seems like the thickened Klein bottle deformation retracts to the front face of it, which in turn is a Klein bottle. If that is the case, then see here. Anyhow, can you define the thickened Klein bottle? My algebraic topology books (Hatcher's and Massey's) doesn't introduce the thickened Klein bottle at all. Is it simply K X [0,1], where K is the Klein bottle?
Yes, it's just Kx[0,1]. (That's how it was defined in the question). I don't know but I would have thought you could thicken anything...?
"it seems like the thickened Klein bottle deformation retracts to the front face of it, which in turn is a Klein bottle"
Well that's what I thought, but that seemed to make the question pointless, so I thought I must be wrong.
If your thickened Klein bottle is simply K X [0,1], it is not difficult to find that it deformation retracts to K, where K is a Klein bottle.
Let Y=K X [0,1]. WLOG, K= K X {0}. Construct a homotopy F:Y X I -->Y such that for all y in Y and x in K, (see here)
F(y,0)=y.
F(y,1) \in K
F(x,1)=x