Intereting Posts

Stirling numbers of the second kind on Multiset
Cauchy Sequence: Multiplication Property
Sum of derivatives of a polynomial
Is this a correct use of the squeeze theorem?
A sequence converges if and only if every subsequence converges?
Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}dx$
Parity of number of factors up to a bound?
$\sqrt x$ is uniformly continuous
Always a double root between “no roots” and “at least one root”?
How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
Power Series proofs
A general formula for the $n$-th derivative of a parametrically defined function
A contradiction involving exponents
Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$
Commutator subgroup and subgroup generated by square.

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I’m having trouble “seeing” this. I’d prefer something that appeals to geometric intuition, but more formal abstract arguments are welcome as well! I want to invoke the unit quaternions here, but I can’t think of a particularly simple way to do it.

If this question is too “soft” I sincerely apologize.

- Orientable Surface Covers Non-Orientable Surface
- The first homology group $ H_1(E(K); Z) $ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian.
- Uniqueness for a covering map lift: is locally connected necessary?
- Wedge product $S^1 \vee S^2$
- Subset of $\mathbb{R}^3$ with an element of finite order in its fundamental group
- Fundamental group of a torus with points removed

- Topology of the space $\mathcal{D}(\Omega)$ of test functions
- connected components equivalence relation
- Consider the “infinite broom”
- Basic facts about ultrafilters and convergence of a sequence along an ultrafilter
- G/H is Hausdorff implies H is closed (General topology, Volume 1 by N. Bourbaki)
- If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.
- “Proof” that $\mathbb{R}^J$ is not normal when $J$ is uncountable
- $C_{c}(X)$ is complete. then implies that $X$ is compact.
- Function which has no fixed points
- Countable Chain Condition for separable spaces?

We discussed here a little while ago the fact that $S^3$ can be described as the result of gluing two solid tori along their boundaries.

To get a geometric reason for the homeomorphism $S^1*S^1\cong S^3$ you can look at how the two constructions are related: notice that two two tori have two central $S^1$s inside them…

(Alternatively: try to see what $S^1*[0,1]$ is, notice that you can describe $S^1$ as two copies of $[0,1]$ identified along the boundaries, and see how you can use this description on one of the two factors of $S^1*S^1$.)

(Another alternative: if you construct $S^1*S^1$ by doing identifications on $S^1\times[0,1]\times S^1$, cut the latter space in two parts $S^1\times[0,1/2]\times S^1$ and $S^1\times[1/2,1]\times S^1$, look at what the identifications do on each half, and *then* glue back the two parts)

- Can $\ln(x)=\lim_{n\to\infty} n\left(x^{\frac1{n}}-1 \right)$ be expressed as an infinite telescoping product?
- How original RS codes and the corresponding BCH codes are related?
- Why would $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ be close to $A^{\dagger}$ when $A$ is with rank deficiency?
- Polynomial division problem
- Integral $\int_1^\infty\frac{dx}{1+2^x+3^x}$
- Easy way to compute logarithms without a calculator?
- Dog Bone Contour Integral
- Why a tesselation of the plane by a convex polygon of 7 or more sides is not possible?
- Expected number of unique items when drawing with replacement
- Is $\delta : \mathcal{S}(\mathbf{R}) \to \mathbf{C}$ continuous with usual seminorm?
- calculate riemann sum of sin to proof limit proposition
- Are there infinitely many primes of the form $12345678901234567890\dots$
- How did Archimedes find the surface area of a sphere?
- Irreducible elements in $\mathbb{Z} $
- What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?