Generalized Poincare conjecture, 4D differentiable case

Question

I recently learnt that the current status of Poincare conjecture:

Top: All true

PL: All true except n=4 being unknown

Diff: True for n=1,2,3,5,6,12; unknown for n=4; False for everything else

Question 1: Is what I learnt correct?

Question 2: Did people try this with the infinite dimensional manifolds? Are fractional dimensional manifolds being defined and studied?

Answer 1

Question 1: From the Wikipedia article on the Generalized Poincare Conjecture, this is correct :

Question 2: Part(a) There are infinite-dimensional manifolds called "Banach Manifolds". However I'm not sure if any work has been done regarding a similar conjecture for these infinite-dimensional manifolds.

Question 2: Part (b) The definition of a topological manifold $M$ of dimension $n$ is a Haursdoff space, together with the added condition that for every point $x \in M$ there exists a neighbourhood $U \subseteq M$ of $x$, hoemomorphic to an open subset of $\mathbb{R}^n$.

Remember $\mathbb{R}^n$ is just the cartesian product of $\mathbb{R}$ taken $n$ times, that is $\mathbb{R}^n = \mathbb{R} \times ... \times \mathbb{R} $ ($n$ times). An element $x$ of $\mathbb{R}^n$, is of the form $x = (x_1, x_2, ...., x_n)$. Hence $\mathbb{R}^{\frac{m}{n}}$, doesn't even make sense, for example what does an element of $\mathbb{R}^{\frac{m}{n}}$ even look like?

Because $\mathbb{R}^{\frac{m}{n}}$ isn't even defined, we can't have a topological manifold $M$ of dimension $\frac{m}{n}$.