Instability of the Smith index under joins and applications to embeddability

Parsa, Salman

doi:10.1090/tran/8698

Instability of the Smith index under joins and applications to embeddability
HTML articles powered by AMS MathViewer

by Salman Parsa PDF

Trans. Amer. Math. Soc. 375 (2022), 7149-7185 Request permission

Abstract:

We say a $d$-dimensional simplicial complex embeds into double dimension if it embeds into the Euclidean space of dimension $2d$. For instance, a graph is planar iff it embeds into double dimension. We study the conditions under which the join of two simplicial complexes embeds into double dimension. Quite unexpectedly, we show that there exist complexes which do not embed into double dimension, however their join embeds into the respective double dimension. We further derive conditions, in terms of the van Kampen obstructions of the two complexes, under which the join will not be embeddable into the double dimension.

Our main tool in this study is the definition of the van Kampen obstruction as a Smith class. We determine the Smith classes of the join of two $\mathbb {Z}_p$-complexes in terms of the Smith classes of the factors. We show that in general the Smith index is not stable under joins. This allows us to prove our embeddability results.

References

Mladen Bestvina, Michael Kapovich, and Bruce Kleiner, Van Kampen’s embedding obstruction for discrete groups, Invent. Math. 150 (2002), no. 2, 219–235. MR 1933584, DOI 10.1007/s00222-002-0246-7
A. Björner, G. M. Ziegler, and J. Matoušek, Using the Borsuk-Ulam theorem: lectures on topological methods in combinatorics and geometry, Universitext, Springer, Berlin Heidelberg, 2008.
P. E. Conner and E. E. Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960), 416–441. MR 163310, DOI 10.1090/S0002-9904-1960-10492-2
F. Effenberger and J. Spreer, simpcomp, a gap toolbox for simplicial complexes, version 2.1.10, https://simpcomp-team.github.io/simpcomp/, June 2019, Refereed GAP package.
R. Flores, Über die Existenz $n$-dimensionaler Komplexe, die nicht in den $R_{2n}$ topologisch einbettbar sind, Ergebnisse eines Mathematischen Kolloquiums, vol. 5, 1932–33, pp. 17–24.
Michael H. Freedman, Vyacheslav S. Krushkal, and Peter Teichner, van Kampen’s embedding obstruction is incomplete for $2$-complexes in $\textbf {R}^4$, Math. Res. Lett. 1 (1994), no. 2, 167–176. MR 1266755, DOI 10.4310/MRL.1994.v1.n2.a4
The GAP Group, GAP – groups, algorithms, and programming, version 4.11.0, 2020.
Rafael Gomes and Gustavo Granja, The cohomological index of free $\Bbb {Z}/p$-actions is not additive with respect to join, Topology Appl. 291 (2021), Paper No. 107612, 8. MR 4208074, DOI 10.1016/j.topol.2021.107612
Branko Grünbaum, Imbeddings of simplicial complexes, Comment. Math. Helv. 44 (1969), 502–513. MR 254851, DOI 10.1007/BF02564551
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Jiří Matoušek, Martin Tancer, and Uli Wagner, Hardness of embedding simplicial complexes in $\Bbb R^d$, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 2, 259–295. MR 2746766, DOI 10.4171/JEMS/252
S. A. Melikhov, The van Kampen obstruction and its relatives, Tr. Mat. Inst. Steklova 266 (2009), no. Geometriya, Topologiya i Matematicheskaya Fizika. II, 149–183; English transl., Proc. Steklov Inst. Math. 266 (2009), no. 1, 142–176. MR 2603266, DOI 10.1134/S0081543809030092
Sergey A. Melikhov and Evgenij V. Shchepin, The telescope approach to embeddability of compacta, arXiv Mathematics e-prints, December 2006, math/0612085.
Karl Menger, Über Plättbare Dreiergraphen und Potenzen nicht plättbarer Graphen, Ergebnisse eines Mathematischen Kolloquiums, vol. 2, 1929, pp. 30–31.
John Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430–436. MR 77932, DOI 10.2307/1970012
Minoru Nakaoka, Cohomology theory of a complex with a transformation of prime period and its applications, J. Inst. Polytech. Osaka City Univ. Ser. A 7 (1956), 51–102. MR 91462
Salman Parsa, On the Smith classes, the van Kampen obstruction and embeddability of $[3]*K$, E-prints, arXiv:2001.06478, January 2020.
Salman Parsa and Arkadiy Skopenkov, On embeddability of joins and their ‘factors’, E-prints, arXiv:2003.12285, March 2020.
K. S. Sarkaria, Kuratowski complexes, Topology 30 (1991), no. 1, 67–76. MR 1081934, DOI 10.1016/0040-9383(91)90034-2
Göran Schild, Some minimal nonembeddable complexes, Topology Appl. 53 (1993), no. 2, 177–185. MR 1247675, DOI 10.1016/0166-8641(93)90135-Z
Arnold Shapiro, Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction, Ann. of Math. (2) 66 (1957), 256–269. MR 89410, DOI 10.2307/1969998
Arkadi Skopenkov, On the Haefliger-Hirsch-Wu invariants for embeddings and immersions, Comment. Math. Helv. 77 (2002), no. 1, 78–124. MR 1898394, DOI 10.1007/s00014-002-8332-4
Arkadiy Skopenkov, Realizability of hypergraphs and Ramsey link theory, Preprint, arXiv:1402.0658, 2014.
Mikhail Skopenkov, Embedding products of graphs into Euclidean spaces, Fund. Math. 179 (2003), no. 3, 191–198. MR 2029321, DOI 10.4064/fm179-3-1
Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554, DOI 10.1007/978-1-4684-9322-1
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258, DOI 10.1515/9781400883875
Brian R. Ummel, Imbedding classes and $n$-minimal complexes, Proc. Amer. Math. Soc. 38 (1973), 201–206. MR 317336, DOI 10.1090/S0002-9939-1973-0317336-9
Brian R. Ummel, The product of nonplanar complexes does not imbed in $4$-space, Trans. Amer. Math. Soc. 242 (1978), 319–328. MR 478130, DOI 10.1090/S0002-9947-1978-0478130-0
E. R. van Kampen, Komplexe in euklidischen Räumen, Abh. Math. Sem. Univ. Hamburg 9 (1933), no. 1, 72–78 (German). MR 3069580, DOI 10.1007/BF02940628
C. Weber, Plongements de polyhèdres dans le domaine métastable, Comment. Math. Helv. 42 (1967), 1–27 (French). MR 238330, DOI 10.1007/BF02564408
Wu Wen-tsün, A theory of imbedding, immersion, and isotopy of polytopes in a euclidean space, Science Press, Peking, 1965. MR 0215305