From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4
Winter Braids VIII (Marseille, 2018), Winter Braids Lecture Notes (2018), Talk no. 4, 19 p.

These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan “Trisections are to 4–manifolds as Heegaard splittings are to 3–manifolds”, focusing on and clarifying the distinction between three ways of thinking of things: the basic definitions as decompositions of manifolds, the Morse theoretic perspective and descriptions in terms of diagrams. We also lay out these themes in two important relative settings: 4–manifolds with boundary and 4–manifolds with embedded 2–dimensional submanifolds.

DOI: 10.5802/wbln.24
Gay, David T 1

1 Euclid Lab 160 Milledge Terrace Athens, GA 30606 Department of Mathematics University of Georgia Athens, GA 30602
@article{WBLN_2018__5__A4_0,
     author = {Gay, David T},
     title = {From {Heegaard} splittings to trisections; porting $3$-dimensional ideas to dimension $4$},
     booktitle = {Winter Braids VIII (Marseille, 2018)},
     series = {Winter Braids Lecture Notes},
     note = {talk:4},
     pages = {1--19},
     publisher = {Winter Braids School},
     year = {2018},
     doi = {10.5802/wbln.24},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/wbln.24/}
}
TY  - JOUR
AU  - Gay, David T
TI  - From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$
BT  - Winter Braids VIII (Marseille, 2018)
AU  - Collectif
T3  - Winter Braids Lecture Notes
N1  - talk:4
PY  - 2018
SP  - 1
EP  - 19
PB  - Winter Braids School
UR  - http://www.numdam.org/articles/10.5802/wbln.24/
DO  - 10.5802/wbln.24
LA  - en
ID  - WBLN_2018__5__A4_0
ER  - 
%0 Journal Article
%A Gay, David T
%T From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$
%B Winter Braids VIII (Marseille, 2018)
%A Collectif
%S Winter Braids Lecture Notes
%Z talk:4
%D 2018
%P 1-19
%I Winter Braids School
%U http://www.numdam.org/articles/10.5802/wbln.24/
%R 10.5802/wbln.24
%G en
%F WBLN_2018__5__A4_0
Gay, David T. From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$, in Winter Braids VIII (Marseille, 2018), Winter Braids Lecture Notes (2018), Talk no. 4, 19 p. doi : 10.5802/wbln.24. http://www.numdam.org/articles/10.5802/wbln.24/

[1] Baykur, R. İnanç; Saeki, Osamu Simplified broken Lefschetz fibrations and trisections of 4-manifolds, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 43, pp. 10894-10900 http://www.pnas.org/content/115/43/10894 | arXiv | DOI | MR | Zbl

[2] Castro, Nickolas A. Relative trisections of smooth 4-manifolds with boundary, Ph. D. Thesis, The University of Georgia (2016)

[3] Castro, Nickolas A.; Gay, David T.; Pinzón-Caicedo, Juanita Diagrams for relative trisections, Pacific J. Math., Volume 294 (2018) no. 2, pp. 275-305 | DOI | MR | Zbl

[4] Gay, David; Kirby, Robion Trisecting 4-manifolds, Geom. Topol., Volume 20 (2016) no. 6, pp. 3097-3132 | DOI | MR | Zbl

[5] Gay, David; Meier, Jeffrey Doubly pointed trisection diagrams and surgery on 2-knots, 2018 | arXiv

[6] Juhász, András Holomorphic discs and sutured manifolds, Algebr. Geom. Topol., Volume 6 (2006), pp. 1429-1457 | DOI | MR | Zbl

[7] Kirby, Robion; Thompson, Abigail A new invariant of 4-manifolds, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 43, pp. 10857-10860 http://www.pnas.org/content/115/43/10857 | arXiv | DOI | MR | Zbl

[8] Lambert-Cole, Peter Bridge trisections in ℂℙ 2 and the Thom conjecture, 2018 | arXiv

[9] Lambert-Cole, Peter; Meier, Jeffrey Bridge trisections in rational surfaces, 2018 | arXiv

[10] Laudenbach, François A proof of Reidemeister-Singer’s theorem by Cerf’s methods, Ann. Fac. Sci. Toulouse Math. (6), Volume 23 (2014) no. 1, pp. 197-221 | DOI | Numdam | MR | Zbl

[11] Laudenbach, François; Poénaru, Valentin A note on 4-dimensional handlebodies, Bull. Soc. Math. France, Volume 100 (1972), pp. 337-344 | DOI | Numdam | MR | Zbl

[12] Meier, Jeffrey Trisecting surfaces filling transverse links (in preparation)

[13] Meier, Jeffrey; Zupan, Alexander Bridge trisections of knotted surfaces in S 4 , Trans. Amer. Math. Soc., Volume 369 (2017) no. 10, pp. 7343-7386 | DOI | MR | Zbl

[14] Meier, Jeffrey; Zupan, Alexander Bridge trisections of knotted surfaces in 4-manifolds, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 43, pp. 10880-10886 http://www.pnas.org/content/115/43/10880 | arXiv | DOI | MR | Zbl

[15] Reidemeister, Kurt Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg, Volume 9 (1933) no. 1, pp. 189-194 | DOI | MR | Zbl

[16] Saltz, Adam Invariants of knotted surfaces from link homology and bridge trisections, 2018 | arXiv

[17] Singer, James Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc., Volume 35 (1933) no. 1, pp. 88-111 | DOI | MR | Zbl

Cited by Sources: