I'm having trouble building the actual deformation retractions, although I understand the concepts behind them, homotopies, etc.
For example, when constructing a deformation retraction for $\mathbb{R}^n-\{0\}$ to $S^{n-1}$, I found that you could define the mapping $F(x,t) = (\frac{x_1}{t||x||+(1-t)},…,\frac{x_n}{t||x||+(1-t)})$.
However, I still don't see how one thought of that in the first place.
I get the idea of turning the $x_n$'s into unit vectors, but I don't understand the intuition behind the $+(1-t)$, etc.
Anyone want to give some advice on how you approach constructing such a family of functions?
In terms of an actual problem, I'm trying to construct a def. retraction of $T_2-\{p\}$ onto a graph with 2 circles intersecting in a point (the longitude/meridian circles of the torus). I understand why this is possible, but my intuition fails to construct the actual def. retraction.
Best Answer
The easiest deformation retracts to think of come from straight line homotopies. Note that the retract of $\mathbb{R}^n \setminus 0$ to $S^{n-1}$ given by Dylan Moreland is of this variety.
In this vein we can also answer your question. $T_2\setminus\{p\}$ is given by the quotient of the unit square with the point $(1/2,1/2)$ removed obtained by identifying opposite sides of the boundary. Try to show that the straight line homotopy of the unit square minus a point to its boundary induces in the quotient a deformation retract of $T_2 \setminus \{p\}$ to the wedge of two circles.