Looking for a "Guide for the Perplexed by Low-dimensional Topology"

Question

The following excerpt is from pp. 56-57 of Loring Tu's (so far very enjoyable) textbook An Introduction to Manifolds (2nd ed.):

One of the most surprising achievements in topology was John Milnor’s discovery [...] in 1956 of exotic $7$-spheres, smooth manifolds homeomorphic but not diffeomorphic to the standard $7$-sphere. In 1963, Michel Kervaire and John Milnor [...] determined that there are exactly 28 nondiffeomorphic differentiable structures on $S^7$.

It is known that in dimensions $< 4$ every topological manifold has a unique differentiable structure and in dimensions $> 4$ every compact topological manifold has a finite number of differentiable structures. Dimension 4 is a mystery. It is not known whether $S^4$ has a finite or infinite number of differentiable structures. The statement that $S^4$ has a unique differentiable structure is called the smooth Poincaré conjecture. As of this writing in 2010, the conjecture is still open.

There are topological manifolds with no differentiable structure. Michel Kervaire was the first to construct an example [...].

For a layperson like myself these are unsettling revelations. In fact, the mere existence of a field of research devoted specifically to low-dimensional topology is already unsettling enough¹.

What accounts for this distinction between low- and high-dimensional topology? What's up with dimension 4? Is $\mathbb{R}^4$ (or $\mathbb{C}^4$ or $\mathbb{H}^4$) topologically exceptional as well?

I have searched online, without success, for a textbook that addresses itself to these questions. Any suggestions?

Alternatively, is there a guide to the proofs of the items mentioned in the excerpt above (or at least of the ones about the number of differentiable structures on $4$-manifolds)?

Either way, I'm interested in a textbook-like treatment (with proofs) aimed at someone whose mathematical level is roughly that of Tu's textbook².

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_{¹Then again, there's this business with the quintic, which I don't yet understand... I wonder if there's any known connection between these two unexpected "dividing lines" in $\mathbb{N}$.}

_{²Tu's textbook, true to its promise to stick to the essentials of its subject, does not elaborate on any of the quoted findings.}

Answer 1

I strongly recommend Scorpan's The Wild World of 4-manifolds. As the title suggests, it's mainly centered on dimension 4, but in its first part, it does a superb job at explaining what is special about low dimensions.