- #1
barbutzo
- 6
- 0
Hello all,
I have a question I'm having a hard time with in an introductory Algebraic Topology course:
Take two handlebodies of equal genus g in S^3 and identify their boundaries by the identity mapping. What is the fundamental group of the resulting space M?
Now, I know you can glue two handlebodies of equal genus to create S^3 itself - but that's gluing is not done by the identity mapping.
Intuitively, it would seem I would get the free group with g generators - it's as if I'm identifying the generators of the fundamental groups of both handlebodies. Thing is, when I'm trying to formalize that notion using the Seifert-Van-Kampen theorem - it doesn't turn out right. If I'm trying to use SVK then after identifying the boundaries, the two handlebodies cover the space, and their intersection is a connected sum of tori. The induced homomorphism from their intersection to each of the handlebodies maps all elements of the intersections fundamental group to e, as they are all equivalent to a point once you can move the loops through the solid handlebody. Now, since the fundamental group of each of the handlebodies is the free group with g generators, it follows from SVK that the fundamental group of M would be the free group with 2g generators (just the free product of both of their fundamental groups, seeing as the intersection didn't create any relations), which, to me, seems twice as much as is correct :)
Any ideas where I'm going wrong?
I have a question I'm having a hard time with in an introductory Algebraic Topology course:
Take two handlebodies of equal genus g in S^3 and identify their boundaries by the identity mapping. What is the fundamental group of the resulting space M?
Now, I know you can glue two handlebodies of equal genus to create S^3 itself - but that's gluing is not done by the identity mapping.
Intuitively, it would seem I would get the free group with g generators - it's as if I'm identifying the generators of the fundamental groups of both handlebodies. Thing is, when I'm trying to formalize that notion using the Seifert-Van-Kampen theorem - it doesn't turn out right. If I'm trying to use SVK then after identifying the boundaries, the two handlebodies cover the space, and their intersection is a connected sum of tori. The induced homomorphism from their intersection to each of the handlebodies maps all elements of the intersections fundamental group to e, as they are all equivalent to a point once you can move the loops through the solid handlebody. Now, since the fundamental group of each of the handlebodies is the free group with g generators, it follows from SVK that the fundamental group of M would be the free group with 2g generators (just the free product of both of their fundamental groups, seeing as the intersection didn't create any relations), which, to me, seems twice as much as is correct :)
Any ideas where I'm going wrong?