#### Geometry & Topology Monographs, Vol. 7 (2004),

Proceedings of the Casson Fest,

Paper no. 10, pages 235--265.

## Ideal triangulations of 3--manifolds I: spun normal surface theory

### Ensil Kang, J Hyam Rubinstein

**Abstract**.
In this paper, we will compute the dimension of the space of spun and
ordinary normal surfaces in an ideal triangulation of the interior of
a compact 3-manifold with incompressible tori or Klein bottle
components. Spun normal surfaces have been described in unpublished
work of Thurston. We also define a boundary map from spun normal
surface theory to the homology classes of boundary loops of the
3-manifold and prove the boundary map has image of finite index. Spun
normal surfaces give a natural way of representing properly embedded
and immersed essential surfaces in a 3-manifold with tori and Klein
bottle boundary [E Kang, `Normal surfaces in knot complements' (PhD
thesis) and `Normal surfaces in non-compact 3-manifolds',
preprint]. It has been conjectured that every slope in a simple knot
complement can be represented by an immersed essential surface [M
Baker, Ann. Inst. Fourier (Grenoble) 46 (1996) 1443-1449 and (with D
Cooper) Top. Appl. 102 (2000) 239-252]. We finish by studying the
boundary map for the figure-8 knot space and for the Gieseking
manifold, using their natural simplest ideal triangulations. Some
potential applications of the boundary map to the study of boundary
slopes of immersed essential surfaces are discussed.
**Keywords**.
Normal surfaces, 3--manifolds, ideal triangulations

**AMS subject classification**.
Primary: 57M25. Secondary: 57N10.

**E-print:** `arXiv:math.GT/0410541`

Submitted to GT on 8 January 2004.
(Revised 29 March 2004.)
Paper accepted 16 March 2004.
Paper published 20 September 2004.

Notes on file formats
Ensil Kang, J Hyam Rubinstein

Department of Mathematics, College of Natural Sciences

Chosun University, Gwangju 501-759, Korea

and

Department of Mathematics and Statistics, The University of Melbourne

Parkville, Victoria 3010, Australia

Email: ekang@chosun.ac.kr, ekang@math.snu.ac.kr, rubin@ms.unimelb.edu.au

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