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?

1$\begingroup$ The Kervaire Invariant problem does not quite fit the parameters of the question, but I feel that it is worth mentioning as a nearexample. $\endgroup$ – James Cranch Sep 30 '14 at 14:23
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.

$\begingroup$ +1. That's a very good example of a large K. Somehow I don't think the OP counts 6,7,8 as "large". I wouldn't either. $\endgroup$ – Johannes Hahn Sep 30 '14 at 22:30

1$\begingroup$ Actually it seems the authors only prove that their example has minimal rank, not minimal dimension. Still a very nice answer. $\endgroup$ – Dan Petersen Oct 1 '14 at 10:02

1$\begingroup$ @DanPetersen  indeed, they also mention a dimension1250 example with rank 74. $\endgroup$ – Adam P. Goucher Dec 23 '17 at 19:15
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 quasiisomorphic (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 (DranishnikovRudiyak) or On nonformal simply connected manifolds (FernandezMunoz) for instance.

3$\begingroup$ As a supplement: More general is every $(k1)$connected compact manifold of dimension $\leq (4k1)$ formal for $k>1$. The result mentioned by Sinan Yalin follows for $k=2$. To get a counterexample in dimension 7, one can (as an alternative to the mentioned papers) realize (via SullivanBargeRealization) the minimal Sullivan algebra $(\bigwedge<v,w,x,y,z>,d)$ where $deg(v)=deg(w)=2$, $deg(x)=deg(y)=deg(z)=3$ and $d(v)=d(w)=0$, $d(x)=v^2$, $d(y)=vw$, $deg(z)=w^2$ as a compact simply connected 7manifold. This result will be nonformal, since $<v,v,w>$ is a nontrivial Massey product. $\endgroup$ – archipelago Sep 30 '14 at 9:34

$\begingroup$ Here is a way to obtain archipelago's example geometrically: Consider the map $S^2\times S^2 \to S^4$ that collapses the complement of a small disk to a point. Then pull back the Hopf fiber bundle $S^3\to S^7 \to S^4$ over this map, and obtain a fiber bundle $S^3 \to X \to S^2\times S^2$. The manifold $X$ is the simply connected nonformal 7manifold whose minimal model is described in the above comment. $\endgroup$ – Aleksandar Milivojevic Jun 9 at 20:26
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.

2$\begingroup$ You might want to mention that $E$ is an open subset of $\mathbb R^n$ (presumably with smooth boundary), and also explain that "of minimal perimeter" refers to the fact that any local variation of $E$ that preserves the local volume of $E$ results in an increase of the perimeter (= area of $\partial E$). (Or should I only require the the first order variation of the perimeter be zero?) $\endgroup$ – André Henriques Sep 30 '14 at 22:15