Intereting Posts

Infinity times $i$
Explanation and Proof of the fourth order Runge-Kutta method
If a graph of $2n$ vertices contains a Hamiltonian cycle, then can we reach every other vertex in $n$ steps?
Polar form of the sum of complex numbers $\operatorname{cis} 75 + \operatorname{cis} 83 + \ldots+ \operatorname{cis} 147$
A couple of GRE questions
Problems with introducing ordered pairs axiomatically
$\langle r\rangle$ maximal $\iff r$ irreducible
solve$\frac{xdx+ydy}{xdy-ydx}=\sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}$
Linear algebra revisited: What do we do when we set a coordinate system?
$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $
Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?
What is the formula for the first Riemann zeta zero?
Approximating continuous functions with polynomials
How can I show this inequality: $-2 \le \cos \theta (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$
The compactness of the unit sphere in finite dimensional normed vector space

This is problem 11 (b) from the first chapter of “Basic Topology” by M.A. Armstrong. The author hasn’t had time to develop many theorems or mathematical machinery, so this problem should be able to be solved by just picturing a series of intermediate steps. It goes

Imagine all the spaces shown in Fig. 1.23 to be made of rubber. For

each pair of spaces X, Y, convince yourself that X can be continuously

deformed into Y.

I’m having trouble with one of the pairs of spaces (the other examples in the problem are unrelated, so I neglected to draw them). The two spaces which I can’t seem to think of a continuous deformation for are

- How to determine space with a given fundamental group.
- Topology of a cone of $\mathbb R\mathbb P^2$.
- Fundamental group of the product of 3-tori minus the diagonal
- $I^2$ does not retract into comb space
- The mapping cylinder of CW complex
- Is the $n^{\text{th}}$ homotopy group isomorphic to $$

The caption for the first picture reads “X = punctured torus”, while the caption for the second picture is “Y = Two cylinders glued together over a square patch”. I’m trying to think of some intermediate steps in the problem. Working backwards, I can see how each of the cylinders in the second picture could be deformed to spheres with two punctures each, but I’m having trouble seeing how the “handle” on the torus is created.

- The product of a cofibration with an identity map is a cofibration
- Is every set in a separable metric space the union of a perfect set and a set that is at most countable?
- Proving that a map is a weak homotopy equivalence
- Prove the image of separable space under continuous function is separable.
- Bases having countable subfamilies which are bases in second countable space
- The mapping cylinder of CW complex
- Homology of a co-h-space manifold
- $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
- Infinite metric space has open set $U$ which is infinite and its complement is infinite
- Why is stable equivalence necessary in topological K-theory?

To give you an alternative way to see this, you certainly know how to obtain a torus by taking a square of paper and gluing the edges together in couples. Now the punctured torus can be obtained the same way by making a hole in the square of paper. If you deform the hole enough and leave only a small strip around the edges, when you glue them together you’ll get you figure.

This is probably clear, given your answer, but just in case a verbal description is helpful for you or others:

I like to think of this the following way: put your hands in the puncture, one on either side, and begin to stretch the puncture around the torus; once you do this, you can imagine that the torus is mostly puncture, with just two small “ribs” left, as in the picture in your answer.

The following intermediate picture, which is taken from a step in a video I found on YouTube uploaded by user esterdalvit, was sufficient to help me see that there is a continuous deformation between spaces X and Y:

Let me share a portable network graphics’ handcrafted, which one has to see it as a drawing but in ${\Bbb{R}}^3$,

- Generating function for $\sum_{k\geq 1} H^{(k)}_n x^ k $
- How to integrate $\cos^2x$?
- Conditional probability containing two random variables
- Why is a matrix of indeterminates diagonalizable?
- Multidimensional Hensel lifting
- Calculate the following: $\lim\limits_{n\to \infty}\int_X n \log(1+(\frac{f}{n})^{\alpha})d\mu$
- Is the group isomorphism $\exp(\alpha x)$ from the group $(\mathbb{R},+)$ to $(\mathbb{R}_{>0},\times)$ unique?
- Infimum over area of certain convex polygons
- Deciding whether two metrics are topologically equivalent in the space $C^1()$
- Intuition behind the Axiom of Choice
- determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.
- Is one structure elementary equivalent to its elementary extension?
- Is 'no solution' the same as 'undefined'?
- Some questions about functions of bounded variation: Jordan's theorem
- Weaker Condition than Differentiability that Implies Continuity