MathOverflow

In Smale's paper "On the Structure of Manifolds", he proves a "relative" version of the h-cobordism theorem:
Theorem (Smale): For $i \in \{1,2\}$, let $M_i$ be a closed oriented smooth $n$-manifold, and let $V_i \subseteq M_i$ be a closed oriented submanifold of dimension $k$. Suppose $k \geq 5$ and that $\pi_1(V_i) = \pi_1(M_i - V_i) = 1$. If the pairs $(M...

Some thought this was not true
The algebra map $\mathbb{F}_2 \rightarrow \mathbb{F}_2[x]/x^2$ is
(1) Such that $\Omega_{ \mathbb{F}_2[x]/x^2/\mathbb{F}_2}$ is one dimensional

For a differentiable real-valued function on $\mathbb{R}^n$, denoting $\partial_i f$ for the $i$th partial derivative, we can define the functional$$T_n(f) = \sum_{i=1}^n \frac{1}{1 + \log(\|\partial_i f\|_2/\|\partial_i f\|_1)} \|\partial_i f\|_2^2...

Let $p$ be a prime such that $2$ is a primitive root of $p$.We want to find a bijective function $f: (\mathbb{Z}/p\mathbb{Z})^× \to (\mathbb{Z}/p\mathbb{Z})^× $ s.t.
$$f(2k) = f(k) + f(f(k)) $$$$f(-k) = -f(k)$$
As far as I could compute $f = Id$ seems to be the only solution....