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Feb
24
comment Lie Groups and Manifolds
The orientation-preserving isometries of the circle is a circle. In general the identity component of the isometry group of an n-torus R^n / L (where L is an n-dimensional lattice in R^n) is the very same n-torus acting on itself by addition. But it would still be nice to find a Riemannian structure on a Lie group G whose full isometry group is the same Lie group G.
Dec
21
comment Which manifolds admit a diffeomorphism of order $n$?
The surface of genus 2 has a diffeomorphism of order 5: Imagine a "carousel" of two parallel closed decagonal 2-disks in 3-space that are connected by 5 equally spaced half-twisted strips between their alternate corresponding edges. This surface has just one boundary component, which we cap off with a 2-disk. It's then easy to check that this is an orientable surface of genus 2, and that rotating the carousel by 2π/5 extends to the 2-disk cap, thereby effecting a diffeomorphism of order 5.
Sep
8
awarded  Necromancer
Jul
7
awarded  Nice Answer
Apr
21
awarded  Good Answer
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15
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10
awarded  Yearling
Sep
30
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Jun
25
awarded  Excavator
Jan
11
revised Examples of common false beliefs in mathematics
it's -> its
Jan
11
comment Are there smooth bodies of constant width?
I took a look at the Fillmore paper, and just before his Corollary to Theorem 2 -- which reads "Corollary. There exists an analytic hypersurface of constant width in E^n having the same group of symmetries as a regular n-simplex." -- he writes "If we imitate the construction of a Reuleux triangle . . .. Thus:" This seems to imply that he is assuming that [the intersection of four balls in 3-space, centered at the vertices of a regular tetrahedron and each with radius = the side-length of the tetrahedron] is a body of constant width. But this is known to be false.
Dec
24
comment $SU(2)$ and the three sphere
It's not so much that you also need det(x) = 1, as that this is exactly the same as saying |a|<sup>2</sup> + |b|<sup>2</sup> = 1.
Sep
14
comment Riemann zeta at even integers
In the functional equation for ζ(s), the term Γ(s) should be Γ(s/2), and the term Γ(1-s) should be Γ((1-s)/2).
Aug
13
comment Irreducible homology 3-spheres that bound smooth contractible manifolds
The usual contractible 2-complex discovered by Bing is called a "House With Two Rooms", and this is not what is depicted in the image above this answer. The 2-complex depicted is not contractible (since a loop around either inner cylinder is a nontrivial 1-cycle, as is easily verified). To obtain the House With Two Rooms, one needs to add two disjoint rectangles IxI to the image, each one intersecting one inner cylinder in an interval, the outer cylinder in an interval, and the 2-complex depicted in its entire (rectangular) boundary circle.
Mar
28
awarded  Yearling
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29
awarded  Yearling
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25
awarded  Nice Answer
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12
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25
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Aug
25
revised Infinite-dimensional complex polynomial or rational Lie algebras and their pseudogroups
changed title