Intereting Posts

If $$ and $$ are relatively prime, then $G=HK$
Proof of a trigonometric expression
Top homology of an oriented, compact, connected smooth manifold with boundary
What's the difference between $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{-d})$?
Points and lines covering them
Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
General Formula for Equidistant Locus of Three Points
Residue of two functions
$Q_8$ is isomorphic to a subgroup of $S_8$ but not to asubgroup of $S_n$ for $n\leq 7$.
The Frobenius Coin Problem
When is an analytic function in $L^2(\Bbb R)$?
Heine-Borel implies Bolzano-Weierstrass theorem
$f'' + f =0$: finding $f$ using power series
Products and Stone-Čech compactification
Higher order corrections to saddle point approximation

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$ and $S^2\to\mathbb CP^{2n+1}\to\mathbb HP^n$.

Question.Does octonionic Hopf fibration $S^7\to S^{15}\to\mathbb OP^1$ give rise to a fibration $\mathbb HP^3\to\mathbb OP^1$?

(On one hand, constructing a map $\mathbb HP^3\to\mathbb OP^1$ *seems*^{(upd-1)} to be easy: take preimage under $S^{15}\to\mathbb HP^3$ and then use octonionic Hopf map. On the other hand, $\mathbb HP^n$ doesn’t admit free involution for $n\ge 2$…)

- The first Stiefel-Whitney class is zero if and only if the bundle is orientable
- contractible and simply connected
- If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.
- Why is the Jordan Curve Theorem not “obvious”?
- fundamental group of $GL^{+}_n(\mathbb{R})$
- Compact $n$-manifold has same integral cohomology as $S^n$?

^{(upd-1)} Wrong! Let $\pi$ be the map $(a,b)\mapsto ab^{-1}$. For octonions $\pi(a,b)$ and $\pi(a\lambda,b\lambda)$ needn’t coincide (because of non-associativity). So this map is not well-defined.

- Visualizing products of $CW$ complexes
- Homeomorphism of the Disk
- Topological spaces admitting an averaging function
- Top Cohomology group of a “punctured” manifold is zero?
- Proof: A loop is null homotopic iff it can be extended to a function of the disk
- Tangent bundle of P^n and Euler exact sequence
- Presentation of the fundamental group of a manifold minus some points
- Sheaf cohomology: what is it and where can I learn it?
- Does homotopy equivalence of pairs $f:(X,A)\to(Y,B)$ induce the homotopy equivalence of pairs $f:(X,\bar A)\to(Y,\bar B)$?
- The union of growing circles is not homeomorphic to wedge sum of circles

It can’t work in the usual way.

Assuming everything works just as in the previous 2 cases, the fiber would be $S^4$. It follows that the bundle $\mathbb{H}P^3\rightarrow\mathbb{O}P^1$ would be orientable since all the pieces are simply connected. Now, one would be able to “fill in the fibers” to get a bundle $D^4\rightarrow X\rightarrow \mathbb{O}P^1$, with $X$ an orientable 13-manifold whose boundary is $\mathbb{H}P^3$. But there is no such manifold.

To see there is no such manifold, recall that the usual proof that there are no free involutions comes from the fact that the first Pontryagin class $p_1(\mathbb{H}P^n) = [2(n+1)-4]u$ where $u$ is a generator of $H^4(\mathbb{H}P^n)$. Thus $p_1\neq 0$ unless $n=1$. Then then using the fact that an involution (or any diffeomorphism) must preserve $p_1$ and applying the Lefshetz fixed point theorem, we find every involution (or diffeomorphism) has a fixed point.

All we need is that $p_1\neq 0$. This, together with the ring structure, implies that the Pontryagin number $\langle p_1^n, u^n\rangle \neq 0$. It follows from a theorem of Thom, that no $\mathbb{H}P^{n}$ is the boundary of an *oriented* manifold. (In fact, if $n$ is even, a similar argument with Stiefel-Whitney classes shows that these manifolds are not the boundary of any manifold, orientable or not).

- Calculating $\int_{\mathcal{S}}x_1^r \, \mathrm dx_1\ldots \, \mathrm dx_n$
- Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area
- The categories Set and Ens
- Proof: $\tan(x)$ is surjective onto $\mathbb R$
- Looking for a smooth curve that is not rational
- An $\varepsilon-\delta$ proof of $\lim_{x\rightarrow c} (2x^ 2 − 3x + 4) = 2c ^2 − 3c + 4$.
- how to find the root of permutation
- Jointly continuous of product in $B(H)$
- Covering projective variety with open sets $U_i$ such that $\pi^{-1}(U_i) \cong U_i \times \Bbb{A}^1$: How to improve geometric intuition?
- Proving $a^ab^bc^c\ge(abc)^{(a+b+c)/3}$ for positive real numbers.
- Prove that the space has a countable dense subset
- conjecture regarding the cosine fixed point
- Completeness of sequence of reals with finitely many nonzero terms
- Why are gauge integrals not more popular?
- Elementary Number Theory.. If a divides..