(I posted this in Particle Physics too)(adsbygoogle = window.adsbygoogle || []).push({});

We call a group G "simply connected" if every curve C(t) in G which is closed (that is, C(0) = C(1) = I) can be continuously deformed into the trivial curve C'(t) = I (where I is the unit element in G). This is formalised saying that, for each closed C(t), there exists a continuous function F: [0, 1]x[0, 1] -> G such that

1) F(0, t) = C(t), for all t

2) F(1, t) = I, for all t

3) F(s, 0) = F(s, 1) = I, for all s

Now, Wald (General Relativity, 1984) says that the Poincare group is not simply connected, beacuse in particular for a rotation of LaTeX Code: 2\\pi about an axis - say z - such a function F does not exist.

My question follows.

Consider the function

F(s, t) := sI + (1 - s)C(t),

where C(t) is the closed curve in Poincare group G associated to a rotation of LaTeX Code: 2\\pi about z, that is,

C(t) =

(1 0 0 0)

(0 cos2\pi t -sin2\pi t 0)

(0 sin2\pi t cos2\pi t 0)

(0 0 0 1)

with t in [0, 1].

This F seems to verify (1)-(3)... where is my mistake?

I thank you for your help, and apologize for the "matrix" above...

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# Is poincare group simply connected?

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