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Parametrized Complexity of Expansion Height

Authors Ulrich Bauer , Abhishek Rathod , Jonathan Spreer

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Author Details

Ulrich Bauer
  • Department of Mathematics, Technical University of Munich (TUM), Boltzmannstr. 3, 85748 Garching b. München, Germany
Abhishek Rathod
  • Department of Mathematics, Technical University of Munich (TUM), Boltzmannstr. 3, 85748 Garching b. München, Germany
Jonathan Spreer
  • School of Mathematics and Statistics, The University of Sydney, NSW 2006 Australia


We want to thank João Paixão for very helpful discussions.

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Ulrich Bauer, Abhishek Rathod, and Jonathan Spreer. Parametrized Complexity of Expansion Height. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 13:1-13:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two simplicial complexes are of the same simple-homotopy type if they can be transformed into each other by a sequence of two basic homotopy equivalences, an elementary collapse and its inverse, an elementary expansion. In this article we consider the following related problem: given a 2-dimensional simplicial complex, is there a simple-homotopy equivalence to a 1-dimensional simplicial complex using at most p expansions? We show that the problem, which we call the erasability expansion height, is W[P]-complete in the natural parameter p.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → W hierarchy
  • Simple-homotopy theory
  • simple-homotopy type
  • parametrized complexity theory
  • simplicial complexes
  • (modified) dunce hat


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  1. J. J. Andrews and M. L. Curtis. Free groups and handlebodies. Proc. Amer. Math. Soc., 16:192-195, 1965. URL:
  2. Jonathan A. Barmak. Algebraic topology of finite topological spaces and applications, volume 2032 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011. URL:
  3. Ulrich Bauer and Abhishek Rathod. Hardness of approximation for Morse matching. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2663-2674. SIAM, Philadelphia, PA, 2019. URL:
  4. Gilbert Baumslag, Alexei G. Myasnikov, and Vladimir Shpilrain. Open problems in combinatorial group theory. Second edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), volume 296 of Contemp. Math., pages 1-38. Amer. Math. Soc., Providence, RI, 2002. URL:
  5. V. V. Borisov. Simple examples of groups with unsolvable word problem. Mat. Zametki, 6:521-532, 1969. Google Scholar
  6. Benjamin A. Burton, Thomas Lewiner, João Paixão, and Jonathan Spreer. Parameterized complexity of discrete Morse theory. ACM Trans. Math. Software, 42(1):Art. 6, 24, 2016. URL:
  7. R. G. Downey and M. R. Fellows. Parameterized complexity. Monographs in Computer Science. Springer-Verlag, New York, 1999. URL:
  8. J. Flum and M. Grohe. Parameterized complexity theory. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 2006. Google Scholar
  9. Michael R. Garey and David S. Johnson. Computers and intractability. W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness, A Series of Books in the Mathematical Sciences. Google Scholar
  10. Masahiro Hachimori. Combinatorics of constructible complexes. PhD thesis, Tokyo University, 2000. URL:
  11. Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. Google Scholar
  12. Cynthia Hog-Angeloni and Wolfgang Metzler, editors. Two-dimensional homotopy and combinatorial group theory, volume 197 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1993. URL:
  13. Sergei Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer, Berlin, second edition, 2007. Google Scholar
  14. James R. Munkres. Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984. Google Scholar
  15. Alexei D. Myasnikov, Alexei G. Myasnikov, and Vladimir Shpilrain. On the Andrews-Curtis equivalence. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), volume 296 of Contemp. Math., pages 183-198. Amer. Math. Soc., Providence, RI, 2002. URL:
  16. J. Paixão and J. Spreer. Random collapsibility and 3-sphere recognition, 2015. Preprint, 18 pages, 6 figures. URL:
  17. Martin Tancer. Recognition of collapsible complexes is NP-complete. Discrete Comput. Geom., 55(1):21-38, 2016. URL:
  18. J. H. C. Whitehead. On incidence matrices, nuclei and homotopy types. Ann. of Math. (2), 42:1197-1239, 1941. URL:
  19. J. H. C. Whitehead. Simple homotopy types. Amer. J. Math., 72:1-57, 1950. URL:
  20. Perrin Wright. Group presentations and formal deformations. Trans. Amer. Math. Soc., 208:161-169, 1975. URL:
  21. E. C. Zeeman. On the dunce hat. Topology, 2:341-358, 1964. URL:
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