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Fundamental Groups

  • #1
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I need to calculate the fundamental group of the following spaces:

[itex] X_1 = \{ (x,y,z) \in \mathbb{R}^3 | x>0 \}[/itex]
[itex] X_2 = \{ (x,y,z) \in \mathbb{R}^3 | x \neq 0 \}[/itex]
[itex] X_3 = \{ (x,y,z) \in \mathbb{R}^3 | (x,y,z) \neq (0,0,0) \}[/itex]
[itex] X_4 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0,y=0, 0 \leqslant z \leqslant 1 \} [/itex]
[itex] X_5 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0, 0 \leqslant y \leqslant 1 \} [/itex]

I fundamentally do not understand what a fundamental group is of how to calculate it. I have read the notes on this but they are so so abstract.
 

Answers and Replies

  • #2
1,357
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What is the definition of a fundamental group?
 
  • #3
1,444
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ok. i think i have answers for the first 3 that im happy with (ive discussed this with a coursemate)

the 4th one i believe the fundamental group is trivial as any path can be contracted to a point. is this correct?

and the 5th one is R£ with a "sheet" removed is was going to say that we can pull the space in around the sheet making a rectangle taht can then be deformed into a circle. this means the fundamental group of X5 is isomoprhic to that of S1 i.e. it is [itex]\mathbb{Z}[/itex]. is this correct?

thanks.
 
  • #4
1,444
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What is the definition of a fundamental group?


thanks. could you take a look at my above post please?
 

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