(Very) High dimensional manifolds Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. However still there are phenomena which occur only up from certain dimensions above 4: For example the famous result of Milnor, which states that each $PL$ manifold of dimension $n$ is in fact smooth provided that $n \leq 7$. My question is the following: Could you give an example of the (reasonable) theorem of the type "each manifold of dimension $n$ have some property $P$ provided that $n \leq K$ (and for $n>K$ there are  counterexamples)," where $K$ is some large number? 
 A: 
Theorem (Simons). Let $E\subset {\mathbb R}^n$ be of minimal perimeter. If $n\le7$, then $\partial E$ is a hyperplane.

If instead $n=2m\ge8$, then Simons provides the example of a minimal surface
$$x_1^2+\cdots+x_m^2=x_{m+1}^2+\cdots+x_{2m}^2$$
whose mean curvature vanishes identically. Bombieri, De Giorgi & Giusti proved that this cone is a minimal surface. This shows that the limit $n\le7$ in the Theorem is sharp.
A: It's not quite as large as 1254, but there is a conjecture about the Yamabe problem which holds small dimensions but fails once the dimension is 25 or more.
When $(M^n, g)$ is a smooth compact Riemannian manifold of dimension
$n \geq 3$ without boundary, the Yamabe Problem consists of finding
a metric $\widetilde{g}$ which has constant scalar curvature and is conformal to g (i.e., satisfies $\widetilde{g} = u^{4/(n-2)} g$ for some smooth function $u$). The Compactness Conjecture stated that the set of solutions to this problem are compact in the $C^2$ topology unless $(M, g)$ is conformally equivalent to the round sphere. In 2008-09, there were a series of papers on this topic where it was shown that conjecture holds for $n \leq 24$ [1], but fails for all dimensions 25 and higher [2,3].
[1] Khuri, M. A.; Marques, F. C.; Schoen, R. M., A compactness theorem for the Yamabe problem, J. Differ. Geom. 81, No. 1, 143-196 (2009). ZBL1162.53029.
[2] Brendle, Simon, Blow-up phenomena for the Yamabe equation, J. Am. Math. Soc. 21, No. 4, 951-979 (2008). ZBL1206.53041.
[3] Brendle, Simon; Marques, Fernando C., Blow-up phenomena for the Yamabe equation. II, J. Differ. Geom. 81, No. 2, 225-250 (2009). ZBL1166.53025.[3]
A: The smallest example of a manifold that is homotopy equivalent to a topological group, but not rationally equivalent to a Lie group has dimension 1254.
A: An important property of the cohomology of manifolds is the formality, which has been at the heart of a lot of work in rational homotopy theory. A simply connected topological space is said to be formal if its Sullivan minimal model is quasi-isomorphic (as an algebra) to its rational cohomology ring. Neisendorfer and Miller proved in their paper Formal and coformal spaces that every simply connected compact manifold of dimension lesser or equal to 6 is formal. There are counterexamples in dimension 7 and above, see for instance the papers Examples of nonformal closed $(k −1)$-connected manifolds of dimensions $4k − 1$ and more (Dranishnikov-Rudiyak) or On nonformal simply connected manifolds (Fernandez-Munoz) for instance.
