Comments 9. Each edge of the polygon gets assigned a letter and an orientation. Notice that the corner point of the rectangle is not a vertex of the cell division. The top and bottom edges are identified as they were for the torus, but the left and right edges are oriented oppositely. Name required.

Topology and its Applications.

27 () North- Let % be a set of n + 1 disjoint simple loops in the boundary of a handlebody. X of genus n, such that. Take a separating, loop in the boundary of a handlebody. This loop bounds two boundary parallel surfaces, each of which is parallel to one of. Given a non-primitive element g of a free group F, which we identify with the fundamental group of a handlebody H, you can ask whether there.

Then g can clearly be geometrically represented in the cover of H corresponding to G. Comment by Saul — March 24, pm Reply.

One can shrink the length dimension of the connecting cylinder to make the one look like the other. A surface's orientability status is a topological invariant. Cameron asks whether this always happens. The flat torus. Can you find such a loop?

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You can check that cell division c has one vertex in the center of the rectangle, two edges one is horizontal, the other is verticaland one face.
But take a closer look, the bug has become mirror-reversed. No loop can bound both a disk and an incompressible higher genus surface because if it did, then the disk would imply a compressing disk for the surface. In other words, given a loop that bounds two distinct incompressible surfaces, can you boundary compress the handlebody in the complement of the loop until the two surfaces are boundary parallel? At each end of an edge is a vertex possibly the same vertex at each endand no two edges intersect in their interiors. |

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Note that. handlebody structure of M.

In particular, when M4 is connected and smooth h-cobordism theorem which implied the proof of the topological. Poincare conjecture. f: ∂W → ∂W, where the loops a, b are mapped to each other by f.

0.

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W f. W. A handlebody Vg of genus g is a 3-manifold bounded by a closed . set of the fundamental group of π1(S, p) by adding loops in the . [McC85] D. McCullough, Twist groups of compact 3-manifolds, Topology 24 (), no.

A surface is a closed, bounded, and connected topological 2-manifold.

At each end of an edge is a vertex possibly the same vertex at each endand no two edges intersect in their interiors. Comment by Henry W — January 9, pm Reply. Thus, for us, a surface is a compact, connected 2-manifold.

## Relators, simple loops and finite covers. Low Dimensional Topology

This manifold is not connected either. Comment by Henry Wilton — February 19, pm Reply.

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TrackBack URI. For instance, as a ball is inflated its volume, curvature, and surface area change; these are geometric properties. But take a closer look, the bug has become mirror-reversed. Leave a Reply Cancel reply Enter your comment here We actually ask whether something stronger is true — rather than asking whether just one lift of g can be made simple, we want the total collection of lifts of g to be embedded. |

A few handlebody surfaces are pictured in Figure In both cases, there are two topological defects for each ring of the handlebody.

or a −1/2 winding number defect loop attached closely to the inner edge of the. Topology and its Applications · Volume 27, Issue 3, DecemberOn primitive sets of loops in the boundary of a handlebody.

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You are commenting using your WordPress. The Klein bottle looks a lot like the torus, but there's a twist. We note that another topological term, compactness, is often used when discussing surfaces. You are commenting using your WordPress. Email required Address never made public.

## Loops bounding incompressible surfaces in handlebodies Low Dimensional Topology

Furthermore, this loop cannot be continuously contracted to a point while staying on the surface.

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If you do happen to find a simple representative then you can glue a 2-handle to H along this curve. The entire boundary of the 2-cell is attached to the single 0-cell, thus creating a well-known surface, the 2-sphere.
This should mean that the homotopy to the boundary descends and hence the original thing was homotopic into the boundary… well, all this may be bogus, but the answer is surely no. This loop bounds two boundary parallel surfaces, each of which is parallel to one of the complementary components in the boundary. If we can find such a finite cover of H in which there is a simple closed curve that projects to g then the fundamental group of the manifold it defines which is a subgroup of the original 1-relator group contains a surface subgroup. Just make sure that all proper cyclic subwords of g are non-trivial in your chosen finite quotient… Comment by Nathan Dunfield — January 8, pm Reply. To count the edges, observe that four edges form the inner diamond, and one edge leaves each vertex of the diamond, for a total of eight edges. |

Comment by Henry Wilton — February 19, pm Reply. Comment by Henry W — January 9, pm Reply.

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