Intereting Posts

When does $V=L$ becomes inconsistent?
Group of order 24 with no element of order 6 is isomorphic to $S_4$
questions about extremal epimorphisms in category theory
Is there an infinite countable $\sigma$-algebra on an uncountable set
What is the number of ways to select ten distinct letters from the alphabet $\{a, b, c, \ldots, z\}$, if no two consecutive letters can be selected?
If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$
Volumes of solids of revolution about y-axis?
Chernoff Bounds. Solve the probability
One-sided smooth approximation of Sobolev functions
Mathematical background for TQFT
Finding an example of a discrete-time strict local martingale.
Let $G$ be a group, $a$ and $b$ are non-identity elements of $G$, $ab=b^2a$ …
Looking for a book on Differential Equations *with solutions*
Solve the given Cauchy problem on the bounded interval
How to analyze convergence and sum of a telescopic series? I can't find a generic form

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 $n\ge 2$…)

- How to show that a diagram is a pushout in the category $\text{TOP}$?
- Second Stiefel-Whitney Class of a 3 Manifold
- Functions continuous in each variable
- Presentation of the fundamental group of a manifold minus some points
- Projective space, explicit descriptions of maps.
- Question about $4$-manifolds and intersection forms

^{(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.

- Homology Group of Quotient Space
- Is there a continuous non constant map $\mathbb{R}^2 \to \mathbb{S}^1$?
- Fundamental Group of the Space X
- References for Topology with applications in Engineering, Computer Science, Robotics
- Is there a initial “bordism-like” homology theory?
- composition of covering maps
- Star-shaped domain whose closure is not homeomorphic to $B^n$
- Finding the fundamental group of the complement of a certain graph
- Winding number question.
- Associativity of the smash product on compactly generated spaces

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).

- arithmetic mean of a sequence converges
- Expressing $\Bbb N$ as an infinite union of disjoint infinite subsets.
- Puzzled by $\displaystyle \lim_{x \to – \infty} \sqrt{x^2+x}-x$
- How to exhibit models of set theory
- Question regarding isomorphisms in low rank Lie algebras
- Proving that none of these elements 11, 111, 1111, 11111…can be a perfect square
- Identity in Number Theory Paper
- Why 4 is not a primitive root modulo p for any prime p?
- A improper integral with complex parameter
- Derivation of the Partial Derangement (Rencontres numbers) formula
- How do I show that every group of order 90 is not simple?
- Multiplicative Functions
- Proving that $ f: \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.
- Decidability of a predicate.
- I need help finding a rigorous precalculus textbook