do "projection" maps satisfy the homotopy lifting property?

Question

I suppose I mean "projection" in terms of local coordinates on a manifold/Lie group. I know that projections in the sense of fiber bundles, fibrations, and covering spaces satisfy the homotopy lifting property. In particular, if I have a matrix group, would projection onto some of the matrix coordinates satisfy HLT? I know projection in terms of a cartesian product $B \times E \to B$ satisfies HLT, since that is covered by trivial bundles, but I don't think this is obviously one of the above 3 cases.

Answer 1

In particular, if I have a matrix group, would projection onto some of the coordinates satisfy HLT?

No. The boundary points of the range can have fibers which "glue" the rest of the domain together in a way that creates holes. For example, consider projecting $\mathrm{SU}(2)$ onto any coordinate.

The range is the closed unit disk, where interior points have circular fibers and boundary points have point fibers. (It's kind of like if you (orthogonally) project a sphere onto a plane: its shadow is a closed unit disk, the shadow's circular boundary has point fibers, while the shadow's interior has zero-sphere fibers.) You can visualize how all these fibers glue to make a three-sphere using stereographic projection: take a circle in three-space around a "generalized" circle (line), then fill in the rest of space between them with toral shells, and partition the tori into circles. Notice the circle fibers shrink as they get closer to the point fibers, which makes sense with what we know about $\mathrm{SU}(2)$ (as one coordinate grows in magnitude, along with its catercorner, the other two adjacent coordinates shrink in magnitude, because $|\alpha|^2+|\beta|^2=1$.)

This projection $\mathrm{SU}(2)\to\mathbb{D}$ does not satisfy the HLP. The projection itself lifts to the identity map $\mathrm{SU}(2)\to\mathrm{SU}(2)$, and the projection is homotopic to $\mathrm{SU}(2)\to\{1\}$ which must lift to a map $\mathrm{SU}(2)\to\{A\}$ (where $A=[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}]$ or $\pm[\begin{smallmatrix}0&\!-1\\1&\,0\end{smallmatrix}]$ depending on which coordinate you picked to project to). If this homotopy had a lift, it would effectively contract $\mathrm{SU}(2)\simeq S^3$ to a point (impossible).