M. M. Postnikov in the book "Smooth Manifolds" called a statement

The sphere $S^{n-1}$ isn't a retract of the ball $B^n$.

the "Drum theorem" because for $n=2$ it mean that we can stretch a film over a circle and make a drum.

How the statements are related?

  • 2
    $\begingroup$ Well, if the Disk would retract to the circle, the drum would not drum if you drum it. The material would flow smoothly to it's border $\endgroup$ – Blah Mar 16 '12 at 10:58

The explanation by Blah is spot on: if a disk could retract to its boundary, the drum head would collapse onto the rim of the drum instead of remaining stretched.

M.M. Postnikov liked this illustration enough that he included it in the Topology article of БСЭ. Loose translation:

This fact, essentially the material of elementary geometry, which for $n=2$ is immediately obvious as the possibility to stretch drumhead on a hoop, still has no proof without the methods of algebraic topology.


этот, по существу, элементарно-геометрический и (при $n = 2$) наглядно очевидный факт (физически означающий возможность натянуть на круглый обруч барабан) до сих пор не удалось доказать без привлечения алгебраико-топологических методов.


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