- #1
Bacle
- 662
- 1
I have read that the complement of a handlebody Hg--an n-ball embedded
in R^n with g handles attached to the ball-- is itself a handlebody.
But this seems false, e.g., for a torus H1 ( a 3-ball with one
handle attached to it ) embedded in R^3: the fundamental group
of the complement is Z --the integers--as any loop about the complement
can be shrunk and translated into a generic loop, while the fund. group
of a handlebody is Z^2g. (note: B^1 is the unit interval in R^1 , etc.)
All I can think is that we may be considering the complement in
S^3 instead of in R^3, but I don't see how an extra point (point at
infinity) would make a difference.
I can see how the complement of an n-ball in R^n retracts to an (n-1)-sphere, e.g.,we can
retract R^2-{(0,0)} into S^1, and we can then consider the complement of an attached
handle/torus, then the union of the complements, i.e., R^n-{B^n\/{handle}} by De Morgan
or something like it. Then I guess we could have a reverse, or "sunken" handle in the
complement, which we can push outward into a "standard handle".
Any Ideas?
Thanks.
Any Ideas?
Thanks.
in R^n with g handles attached to the ball-- is itself a handlebody.
But this seems false, e.g., for a torus H1 ( a 3-ball with one
handle attached to it ) embedded in R^3: the fundamental group
of the complement is Z --the integers--as any loop about the complement
can be shrunk and translated into a generic loop, while the fund. group
of a handlebody is Z^2g. (note: B^1 is the unit interval in R^1 , etc.)
All I can think is that we may be considering the complement in
S^3 instead of in R^3, but I don't see how an extra point (point at
infinity) would make a difference.
I can see how the complement of an n-ball in R^n retracts to an (n-1)-sphere, e.g.,we can
retract R^2-{(0,0)} into S^1, and we can then consider the complement of an attached
handle/torus, then the union of the complements, i.e., R^n-{B^n\/{handle}} by De Morgan
or something like it. Then I guess we could have a reverse, or "sunken" handle in the
complement, which we can push outward into a "standard handle".
Any Ideas?
Thanks.
Any Ideas?
Thanks.