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Connected components of upper triangular matrices

  1. Apr 18, 2006 #1
    Hello, i'm working on a problem in topology. I'm supposed to find the number of connected components of the group of 2x2 invertible upper triangular matrices over R which i shall call [itex]B_2[/itex].

    I've tried it a bit, but I don't know for sure if my approach (and answer) is correct.

    Since any homeomorphism preserves connectivity, I consider the trivial homeomorphism of [itex]B_2[/itex] to [itex]\mathbf{R}^3[/itex].

    Since the matrices have to be invertible the determinant is non-zero which means that for matrices (a b | 0 c), ac > 0 or ac < 0. But he piece with positive determinant splits in 2 non connected pieces, {(a,b,c) | a > 0 and c > 0} and {(a,b,c) | a < 0 and c < 0}. The same sort of thing holds for the piece with negative determinant.

    Is it correct to assert from this that [itex]B_2[/itex] has 4 connected components?
  2. jcsd
  3. Apr 18, 2006 #2

    matt grime

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    I think that shows it splits into at least 4 connected components. You've not show that each of those individual pieces is connected.
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