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Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary

  • Biplab Basak ORCID logo EMAIL logo and Manisha Binjola ORCID logo
Published/Copyright: November 26, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 2

Abstract

Let 𝑀 be a connected compact PL 4-manifold with boundary. In this article, we give several lower bounds for regular genus and gem-complexity of the manifold 𝑀. In particular, we prove that if 𝑀 is a connected compact 4-manifold with ℎ boundary components, then its gem-complexity k⁢(M) satisfies the inequalities k⁢(M)≥3⁢χ⁢(M)+7⁢m+7⁢h-10 and k⁢(M)≥k⁢(∂⁡M)+3⁢χ⁢(M)+4⁢m+6⁢h-9, and its regular genus G⁢(M) satisfies the inequalities G⁢(M)≥2⁢χ⁢(M)+3⁢m+2⁢h-4 and G⁢(M)≥G⁢(∂⁡M)+2⁢χ⁢(M)+2⁢m+2⁢h-4, where 𝑚 is the rank of the fundamental group of the manifold 𝑀. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds is also shown for a large class of PL 4-manifolds with boundary.

Keywords: PL 4-manifold with boundary; crystallization; regular genus; gem-complexity
MSC 2010: 57Q15; 57Q05; 57K41; 05C15

Funding source: Department of Science and Technology, Ministry of Science and Technology, India

Award Identifier / Grant number: DST/INSPIRE/04/2017/002471

Funding statement: The first author is supported by DST INSPIRE Faculty Research Grant (DST/INSPIRE/04/2017/002471).

Acknowledgements

The authors would like to thank the anonymous referees for many useful comments and suggestions.

  1. Communicated by: Frederick R. Cohen

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Received: 2020-04-15
Revised: 2020-09-06
Published Online: 2020-11-26
Published in Print: 2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
https://doi.org/10.1515/forum-2020-0093
Keywords for this article
PL 4-manifold with boundary; crystallization; regular genus; gem-complexity

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
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