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# Exotic spheres, or why 4-dimensional space is a crazy place

### by Richard Elwes

*differential topology*has grown up, and revealed just how alien these places are.

### Higher dimensions and hyperspheres

*hypersphere*.

### From geometry to topology to differential topology

*Poincaré conjecture*, provides an elegant answer to this question: it says that the only such shapes are the spheres. This is not true from a geometrical viewpoint, as cubes, pyramids, dodecahedra, and a multidue of other shapes all have no holes. But, of course, to a topologist, all these exciting shapes are nothing more than spheres.

*Plus*). Henri Poincaré's original question concerned the 3-sphere, but in fact exactly the same thing applies in all higher dimensions too. The fact is that, when viewed topologically, spheres are beautifully simple and unique objects in every dimension. However, in 1956 the first evidence arrived that a slight change in perspective would make the story hugely more complicated. When approached through the new subject of differential topology, higher dimensional spaces began to reveal some of their extraordinary secrets.

### Gaps, kinks, and corners

*Julia set*. Its outline is continuous, but nowhere smooth.

*continuous*, meaning that they do not jump or tear, and others which are

*smooth*. Smoothness is a much stronger condition than mere continuity. The same distinction applies to shapes themselves: circles and spheres are examples of smooth shapes, while squares and cubes are not smooth because of their sharp edges and corners. All of these are continuous, however, because their edges do not have any gaps or jumps. (A discontinuous line is one which comes in two separate pieces.) There are even fractal patterns which are continuous everywhere, but not smooth anywhere.

*manifolds*by topologists) could be the same from a topological perspective (in technical terms, be

*homeomorphic*), but not the same from a differential perspective (they are not

*diffeomorphic*)? In other words, can we have two shapes that can be morphed into each other without cutting, but for which the morphing can't be smooth, it requires jerks and jumps? This is certainly difficult to imagine, not least because it never happens in dimensions 1, 2, or 3.

### Exotic spheres

*exotic sphere*, and he went on to find several more in other dimensions. In each case, the result was topologically spherical, but not differentially so. Another way to say the same thing is that the exotic spheres represent ways to impose unusual notions of distance and curvature on the ordinary sphere.

Dimension | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

Number of spheres | 1 | 1 | 1 | ? | 1 | 1 | 28 | 28 | 6 | 992 | 1 | 3 | 2 | 16256 | 2 | 16 | 16 |

*smooth Poincaré*conjecture. In case anyone has got this far and is still not sure, let me make this clear: the smooth Poincaré conjecture is not the same thing as the Poincaré conjecture! Among other differences, the Poincaré conjecture has been proved, but the smooth Poincaré conjecture remains stubbornly open today.

### The weird world of four dimensions

*dodecaplex*. Image crated by Paul Nylander.

^{4}) comes in a variety of flavours. There is the usual flat space, but alongside it are the exotic R

^{4}s. Each of these is topologically identical to ordinary space, but not differentially so. Amazingly, as Clifford Taubes showed in 1987, there are actually infinitely many of these alternative realities. In this respect, the fourth dimension really is an infinitely stranger place than every other domain: for all other dimensions

*n*, there is only ever one version of R

^{n}. Perhaps after all, the fourth dimension is the right mathematical setting for the weird worlds of science fiction writers' imaginations.

### About the author

*Plus*New Writers Award 2006 and has since published articles in

*New Scientist*and

*Daily Telegraph*, and regularly talks about maths in public and on the radio. He is the author of the book

*Maths 1001*, reviewed in

*Plus*, and

*How to build a brain and 34 other really interesting uses of mathematics*(March 2011)

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