Poincaré dodecahedral space

Poincaré dodecahedral space

Poincaré originally conjectured

[

4

]

that a

homology

3-sphere (

http://planetmath.org/HomologySphere

) must be

homeomorphic

to

S

3

. He later found a counterexample based on the group of

rotations

of the

regular dodecahedron

(

http://planetmath.org/RegularPolyhedron

), and restated his conjecture in of the

fundamental group

. (See

[

5

]

). To be accurate, the restatement took the form of a question. However it has always been referred to as Poincaré’s Conjecture.)

This conjecture was one of the

http://www.claymath.org/

Clay Mathematics Institute’s Millennium Problems. It was finally proved by Grisha Perelman (

http://planetmath.org/GrigoriPerelman

) as a
corollary of his on Thurston’s

geometrization conjecture

(

http://planetmath.org/ThurstonsGeometrizationConjecture

). Perelman was awarded the Fields Medal (

http://planetmath.org/FieldsMedal

) for this work, but he

http://news.bbc.co.uk/2/hi/science/nature/5274040.stm

declined the award.
Perelman’s manuscripts can be found at the arXiv:

[

1

]

,

[

2

]

,

[

3

]

.

Here we take a quick look at Poincaré’s example. Let

Γ

be the rotations of the
regular dodecahedron (

http://planetmath.org/RegularPolyhedron

). It is easy to check that

Γ

≅

A

5

. (Indeed,

Γ

permutes transitively (

http://planetmath.org/GroupAction

) the 6 pairs of faces, and the

stabilizer

of any pair induces a

dihedral group

of order (

http://planetmath.org/OrderGroup

) 10.) In particular,

Γ

is

perfect

. Let

P

be the

quotient space

P

=

S

⁢

O

3

⁢

(

ℝ

)

/

Γ

. We check that

P

is a

homology sphere

.

To do this it is easier to work in the

universal cover

S

⁢

U

⁢

(

2

)

of

S

⁢

O

3

⁢

(

ℝ

)

, since

S

⁢

U

⁢

(

2

)

≅

S

3

. The of

Γ

to

S

⁢

U

⁢

(

2

)

will be denoted

Γ

^

. Hence

P

=

S

⁢

U

⁢

(

2

)

/

Γ

^

.

Γ

^

is a nontrivial central of

A

5

by

{

±

I

}

, which means that there is no

splitting

to the surjection

Γ

^

→

Γ

. In fact

A

5

has no nonidentity 2-dimensional

unitary representations

. In particular,

Γ

^

, like

Γ

, is perfect (

http://planetmath.org/PerfectGroup

).

Now

π

1

⁢

(

P

)

≅

Γ

^

, whence

H

1

⁢

(

P

)

=

0

(since it is the

abelianization

of

Γ

^

). By Poincaré

duality

and the

universal coefficient theorem

(

http://planetmath.org/UniversalCoefficentTheorem

),

H

2

⁢

(

P

)

≅

0

as well. Thus,

P

is indeed a homology sphere.

The

dodecahedron

is a fundamental in a tiling of hyperbolic 3-space, and hence

P

can also be realized by gluing the faces of a dodecahedron. Alternatively, Dehn showed how to construct this same example using

surgery

around a trefoil.
Dale Rolfson’s fun book

[

6

]

has more on the surgical view of Poincaré’s example.

References

1

G. Perelman,

http://arxiv.org/abs/math.DG/0211159/

“The entropy formula for the

Ricci flow

and its geometric applications”,

2

G. Perelman,

http://arxiv.org/abs/math.DG/0303109/

“Ricci flow with surgery on three-manifolds”,

3

G. Perelman,

http://arxiv.org/abs/math.DG/0307245/

“Finite extinction time for the

solutions

to the Ricci flow on certain three-manifolds”.

4

H. Poincaré, “Second complément à l’

analysis situs

”, Proceedings of the LMS, 1900.

5

H. Poincaré, “Cinquième complément à l’analysis situs”, Proceedings of the LMS, 1904.

6

D. Rolfson, Knots and Links. Publish or Perish Press, 1976.

Title

Poincaré dodecahedral space

Canonical name

PoincareDodecahedralSpace

Date of creation

2013-03-22 13:56:21

Last modified on

2013-03-22 13:56:21

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

65

Author

Mathprof (13753)

Entry type

Example

Classification

msc 57R60

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