Abstract
We describe an algorithm for computing a finite, and typically small, presentation of the fundamental group of a finite regular CW-space. The algorithm is based on the construction of a discrete vector field on the \(3\)-skeleton of the space. A variant yields the homomorphism of fundamental groups induced by a cellular map of spaces. We illustrate how the algorithm can be used to infer information about the fundamental group \(\pi _1(K)\) of a metric space \(K\) using only a finite point cloud \(X\) sampled from the space. In the special case where \(K\) is a \(d\)-dimensional compact manifold \(K\subset \mathbb R^d\), we consider the closure of the complement of \(K\) in the \(d\)-sphere \(M_K=\overline{\mathbb S^d\!\setminus \!K}\). For a base-point \(x\) in the boundary \(\partial M_K\) of the manifold \(M_K\) one can attempt to determine, from the point cloud \(X\), the induced homomorphism of fundamental groups \(\phi :\pi _1(\partial M_K,x)\rightarrow \pi _1(M_K,x)\) in the category of finitely presented groups. We illustrate a computer implementation for \(K\) a small closed tubular neighbourhood of a tame knot in \(\mathbb R^3\). In this case the homomorphism \(\phi \) is known to be a complete ambient isotopy invariant of the knot. We observe that low-index subgroups of finitely presented groups provide useful invariants of \(\phi \). In particular, the first integral homology of subgroups \(G < \pi _1(M_K)\) of index at most six suffices to distinguish between all prime knots with 11 or fewer crossings (ignoring chirality). We plan to provide formal time estimates for our algorithm and characteristics of a high performance C++ implementation in a subsequent paper. The prototype computer implementation of the present paper has been written in the interpreted gap programming language for computational algebra.
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References
Boone, W.W.: The word problem. Ann. Math. 2(70), 207–265 (1959)
Brendel, P., Dłotko, P., Ellis, G., Juda, M., Mrozek, M.: An algorithm for the fundamental group of a regular cw-space (provisional title). In preparation, (2014)
Bridson, M.R., Tweedale, M.: Deficiency and abelianized deficiency of some virtually free groups. Math. Proc. Camb. Philos. Soc. 143(2), 257–264 (2007)
The CAPD Group. CAPD::RedHom - Reduction homology algorithms, the topological part of CAPD, 2013. http://redhom.ii.uj.edu.pl/ and http://capd.ii.uj.edu.pl/
Cha, J.C., Livingston, C.: KnotInfo: table of knot invariants. KnotInfo web page, 2013. http://www.indiana.edu/~knotinfo
Cohen, M.M.: A Course in Simple Homotopy Theory. Graduate Texts in Mathematics. Springer, New York (1973)
Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)
Ellis, G.: HAP - Homological algebra programming, Version 1.10.13, 2013. http://www.gap-system.org/Packages/hap.html
Ellis, G., Hegarty, F.: Computational homotopy of finite regular cw-spaces. J. Homotopy Relat. Struct. 9, 25–54 (2014). doi:10.1007/s40062-013-0029-4
Erickson, J., Whittlesey, K.: Greedy optimal homotopy and homology generators. In: Proceedings of the sixteenth annual ACM-SIAM symposium on discrete algorithms, pages 1038–1046 (electronic). ACM, New York, (2005)
Forman, R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)
Forman, R.: A user’s guide to discrete Morse theory. Sém. Lothar. Combin., 48:Art. B48c, 35, (2002)
The GAP Group. GAP - Groups, algorithms, and programming, Version 4.5.6. http://www.gap-system.org (2013)
Geoghegan, R.: Topological Methods in Group Theory, volume 243 of Graduate Texts in Mathematics. Springer, New York (2008)
González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2–3), 245–263 (2005)
Gordon, C.McA, Luecke, J.: Knots are determined by their complements. J. Am. Math. Soc. 2(2), 371–415 (1989)
Harker, S., Mischaikow, K., Mrozek, M., Nanda, V.: Discrete Morse theoretic algorithms for computing homology of complexes and maps. Found. Comput. Math. 14, 151–184 (2014). doi:10.1007/s10208-013-9145-0
Harker, S., Mischaikow, K., Mrozek, M., Nanda, V., Wagner, H., Juda, M., Dłotko, P.: The efficiency of a homology algorithm based on discrete morse theory and coreductions. Proceedings of the 3rd International Workshop on Computational Topology in Image Context, Chipiona, Spain, November 2010 1, 41–47 (2010)
Jones, D.W.: A general theory of polyhedral sets and the corresponding \(T\)-complexes. Dissertationes Math. (Rozprawy Mat.) 266, 110 (1988)
Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology, volume 157 of Applied Mathematical Sciences. Springer, New York (2004)
Kaczynski, T., Mrozek, M., Ślusarek, M.: Homology computation by reduction of chain complexes. Comput. Math. Appl. 34(4), 59–70 (1998)
Kim, J., Jin, M., Zhou, Q.-Y., Luo, F.: Computing fundamental group of general 3-manifold. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Porikli, F., Peters, J., Klosowski, J., Arns, L., Chun, Y., Rhyne, T.-M., Monroe, L. (eds.) Advances in Visual Computing, volume 5358 of Lecture Notes in Computer Science, pp. 965–974. Springer, Berlin (2008)
Knot Atlas. Knot Atlas. http://katlas.math.toronto.edu/wiki/ (2013)
Letscher, D.: On persistent homotopy, knotted complexes and the Alexander module. In: Proceedings of the 3rd innovations in theoretical computer science conference, ITCS ’12, pages 428–441, New York, NY, USA, ACM (2012)
Massey, W.S.: A basic course in algebraic topology, volume 127 of Graduate Texts in Mathematics. Springer, New York (1991)
Novikov, P.S.: Ob algoritmičeskoĭ nerazrešimosti problemy toždestva slov v teorii grupp. Trudy Mat. Inst. im. Steklov. no. 44. Izdat. Akad. Nauk SSSR, Moscow, (1955)
Nureki, D., Watanabe, K., Fukai, S., Ishii, R., Endo, Y., Hori, H., Yokoyama, S.: Deep knot structure for construction of active site and cofactor binding site of trna modification enzyme. Structure, 12, 593. http://www.rcsb.org/pdb/explore/explore.do?structureId=1V2X (2004)
Palmieri, J.H., et al.: Finite simplicial complexes. Sage v5.10. http://www.sagemath.org/doc/reference/homology/sage/homology/simplicial_complex.html (2009)
Rees, S., Soicher, L.H.: An algorithmic approach to fundamental groups and covers of combinatorial cell complexes. J. Symb. Comput. 29(1), 59–77 (2000)
Spanier, E.H.: Algebraic Topology, Corrected Reprint of the 1966 Original. Springer, New York (1981)
Stein, W.A., et al.: Sage mathematics software (Version 5.10). The Sage development team. http://www.sagemath.org (2013)
Waldhausen, F.: On irreducible \(3\)-manifolds which are sufficiently large. Ann. Math. (2) 87, 56–88 (1968)
Whitehead, J.H.C.: Combinatorial homotopy. I. Bull. Am. Math. Soc. 55, 213–245 (1949)
Whitehead, J.H.C.: Combinatorial homotopy. II. Bull. Am. Math. Soc. 55, 453–496 (1949)
Whitehead, J.H.C.: Simple homotopy types. Am. J. Math. 72, 1–57 (1950)
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P.D. is supported by the Grant DARPA: FA9550-12-1-0416 and AFOSR: FA9550-14-1-0012. G.E. was partially supported by the European Science Foundation network on Applied and Computational Algebraic Topology and by Polish MNSzW, Grant N N201 419639. G.E. thanks the IST, Austria for its hospitality during the writing of this paper. M.M. was partially supported by Polish MNSzW, Grant N N201 419639.
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Brendel, P., Dłotko, P., Ellis, G. et al. Computing fundamental groups from point clouds. AAECC 26, 27–48 (2015). https://doi.org/10.1007/s00200-014-0244-1
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DOI: https://doi.org/10.1007/s00200-014-0244-1