Homotopy groups of suspended classifying spaces: An experimental approach

Abstract

When the results of a computer program are compared to some theorems proved on a theoretical basis three situations can occur: there can be an agreement between both approaches, the computer program can obtain calculations not covered by the theorems, or a discrepancy can be found between both methods. In this paper we report on a work where the three above mentioned situations happen. We have enhanced the Computer Algebra called Kenzo to deal with the computation of homotopy groups of suspended classifying spaces, a problem tackled by Mikhailov and Wu in a paper published in the journal Algebraic and Geometric Topology. Our experimental approach, based on completely different methods from those by Mikhailov and Wu, has allowed us in particular to detect an error in one of their published theorems.

Effective homology of filtered digital images

“…Kenzo has made it possible to detect an error in a theorem published in [12], where some theoretical reasonings are used to deduce that the fourth homotopy group of the suspended classifying space of the fourth alternating group A 4 , π 4 (ΣK(A 4 , 1)), is equal to Z 4 ; Kenzo's calculations have showed that the correct result (as later confirmed by the authors of [12]) is Z 12 . See [16] for details on these calculations. Moreover, in [15] Kenzo has been used to deduce the correct relation between persistent homology and spectral sequences and detect an error in [5]: the so called "Spectral sequence theorem" [5, p. 171] includes a formula which is not correct (see [15] for details).…”

Certified symbolic manipulation

“…Kenzo is a Common Lisp program, created by F. Sergeraert, that can deal with infinite dimensional spaces, and is able to compute results that cannot be determined by any other means (theoretical or computational). In [13] a theorem corrected thanks to Kenzo is presented, together with other results computed with Kenzo which seem out of reach for any other method.…”

Verifying an Algorithm Computing Discrete Vector Fields for Digital Imaging

“…Kenzo [10] is a Computer Algebra System devoted to Algebraic Topology which was developed by F. Sergeraert. This system has computed some homology and homotopy groups which cannot be easily obtained by theoretical or computational means; some examples can be seen in [23]. Therefore, in this situation, it makes sense to analyze the Kenzo programs in order to ensure the correctness of the mathematical results which are obtained thanks to it.…”