Inducing a map on homology from a correspondence

Harker, Shaun; Kokubu, Hiroshi; Mischaikow, Konstantin; Pilarczyk, Paweł

doi:10.1090/proc/12812

Inducing a map on homology from a correspondence
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by Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow and Paweł Pilarczyk

Proc. Amer. Math. Soc. 144 (2016), 1787-1801

DOI: https://doi.org/10.1090/proc/12812

Published electronically: August 12, 2015

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Abstract:

We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.

References

Zin Arai, William Kalies, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka, and PawełPilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst. 8 (2009), no. 3, 757–789. MR 2533624, DOI 10.1137/080734935
Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
Felix E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301. MR 229101, DOI 10.1007/BF01350721
Justin Bush, Marcio Gameiro, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Ippei Obayashi, and Paweł Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, CHAOS 22 (2012), no. 4, 047508, DOI 10.1063/1.4767672.
Justin Bush and Konstantin Mischaikow, Coarse dynamics for coarse modeling: An example from population biology, Entropy 16 (2014), no. 6, 3379–3400.
CAPD, Computer Assisted Proofs in Dynamics, http://capd.ii.uj.edu.pl/.
CHomP, Computational Homology Project. Homology Software, http://chomp.rutgers.edu/Software/Homology.html.
Samuel Eilenberg and Deane Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214–222. MR 16676, DOI 10.2307/2371832
Shaun Harker, Generalized homology of maps. Supplemental materials, ttp://chomp.rutgers.edu/Archives/Computational_Homology/GeneralizedHomologyOfMaps/SupplementalMaterials.html.
Shaun Harker, Konstantin Mischaikow, Marian Mrozek, and Vidit Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math. 14 (2014), no. 1, 151–184. MR 3160710, DOI 10.1007/s10208-013-9145-0
M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), no. 1, 69–77. MR 422932
Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek, Computational homology, Applied Mathematical Sciences, vol. 157, Springer-Verlag, New York, 2004. MR 2028588, DOI 10.1007/b97315
Diána H. Knipl, PawełPilarczyk, and Gergely Röst, Rich bifurcation structure in a two-patch vaccination model, SIAM J. Appl. Dyn. Syst. 14 (2015), no. 2, 980–1017. MR 3355763, DOI 10.1137/140993934
Eduardo Liz and PawełPilarczyk, Global dynamics in a stage-structured discrete-time population model with harvesting, J. Theoret. Biol. 297 (2012), 148–165. MR 2899025, DOI 10.1016/j.jtbi.2011.12.012
Konstantin Mischaikow and Marian Mrozek, Conley index, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 393–460. MR 1901060, DOI 10.1016/S1874-575X(02)80030-3
Konstantin Mischaikow, Marian Mrozek, and Pawel Pilarczyk, Graph approach to the computation of the homology of continuous maps, Found. Comput. Math. 5 (2005), no. 2, 199–229. MR 2149416, DOI 10.1007/s10208-004-0125-2
PawełPilarczyk and Pedro Real, Computation of cubical homology, cohomology, and (co)homological operations via chain contraction, Adv. Comput. Math. 41 (2015), no. 1, 253–275. MR 3311480, DOI 10.1007/s10444-014-9356-1
L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), no. 1, 454–472 (German). MR 1512371, DOI 10.1007/BF01447877