## Final Activity Report Summary - COHO (Computational Homological Algebra)

The homology of a topological space was introduced by Poincare at the end of the nineteenth century and is one of the best understood topological invariants. The more intricate homotopy groups were introduced by Hurewicz in the 1930s as generalisations of Poincare's fundamental group. Basic mathematical results on the relationship between homology and homotopy groups were soon established. The relationship is quite subtle and the homology of groups investigates one aspect of it, namely the extent to which homology of a space depends on the fundamental group of the space. One defines the homology of a discrete group G to be the homology of any classifying space for the group (i.e. any connected space X with fundamental group p_1X = G and higher homotopy groups pi_nX = 0 for n > 1). This definition leads to an interesting interplay between topology, geometry, homological algebra, group theory, number theory and other areas of mathematics.

The COHO project developed practical algorithms for computing the homology of spaces and groups and implemented these algorithms in the form of a publicly available software package. Much of this Homological Algebra Package (HAP) has now been refereed and accepted as a component of the international computer algebra system GAP. The algorithms were initially aimed at spaces arising theoretically as classifying spaces of groups (and related structures such as Lie algebras and crossed modules).

Towards the end of the project the focus broadened to include spaces modelling real-life data from medical images, bioinformatics and dynamical systems. Some example computations by the COHO project:

1. David Green (Jena), Simon King (Marie Curie Fellow at Galway) and G.Ellis (Galway) have computed the mod-2 cohomology ring H^*(B(Co3),Z_2) of the classifying space of the third Conway group and thus confirmed a conjecture of David Benson that the ring is Cohen-Macaulay. The computation has been accepted for publication in Algebraic and Geometric Topology.

2. Mathieu Dutour (Marie Curie Fellow at Galway) and G. Ellis (Galway) have computed the low-dimensiona integral homology of all Mathieu finite simple groups. [M.Dutour & G.Ellis, Wythoff polytopes and low-dimensional homology of Mathieu groups, J. Algebra 322 (2009), no. 11, 4143-4150.]

3. Mathieu Dutour (Marie Curie Fellow at Galway) and G. Ellis (Galway) have computed the low-dimensiona integral homology of infinite arithmetic groups such as PSL(4,Z) and PSL(3,Z[i]). This work is currently being written up.

4. Graham Ellis (Galway) and Paul Smith (Marie Curie Fellow at Galway) have used new techniques to recompute (and thus verify) the mod 2 cohomology rings of the groups of order 64. [G.Ellis & P.Smith, Computing group cohomology rings from the Lyndon-Hochschild-Serre spectral sequence. J. Symbolic Computation. In press, DOI:10.1016/j.jsc.2010.09.001] 4. Jie Wu (Singapore) and Roman Mikhailov (Moscow) have used the COHO software to obtain new results on homotopy groups of suspensions. ["On homotopy groups of the suspended classifying spaces", [J.Wu & R.Mikhailov, Algebraic and Geometric Topology 10 (2010), 565-625].

5. Bettina Eick, Heiko Dietrich and D¨orte Feichtenshlager ["Investigating p-groups by coclass with GAP", AMS Contemporary Mathematics 470 (2007), 45-63] have used the COHO software to investigate common cohomological properties of primepower groups of common coclass. 5. The COHO software has helped a number of other distinguished mathematicians with their research. For instance: a) It recently provided Alejandro Adem (British Columbia) with low-dimensional integral cohomology computations on symmetric groups of degree <=12; b) Marc Roeder (Marie Curie Fellow at Galway) provided Bartosz Putrycz (Gdansk/Leuven) with the integral cohomology of Hantzsche-Wendt (Bieberbach) groups of dimension <=8. c) Craig Westerland (Wisconsin) is using the COHO software to compute the cohomology of braid groups with certain twisted integral coefficients.

6. COHO software is being used by a PhD student at Galway to help with the automatic interpretation of X-Ray/Ultrasound images arising in the radiotherapy treatment of prostate cancer.

The COHO project developed practical algorithms for computing the homology of spaces and groups and implemented these algorithms in the form of a publicly available software package. Much of this Homological Algebra Package (HAP) has now been refereed and accepted as a component of the international computer algebra system GAP. The algorithms were initially aimed at spaces arising theoretically as classifying spaces of groups (and related structures such as Lie algebras and crossed modules).

Towards the end of the project the focus broadened to include spaces modelling real-life data from medical images, bioinformatics and dynamical systems. Some example computations by the COHO project:

1. David Green (Jena), Simon King (Marie Curie Fellow at Galway) and G.Ellis (Galway) have computed the mod-2 cohomology ring H^*(B(Co3),Z_2) of the classifying space of the third Conway group and thus confirmed a conjecture of David Benson that the ring is Cohen-Macaulay. The computation has been accepted for publication in Algebraic and Geometric Topology.

2. Mathieu Dutour (Marie Curie Fellow at Galway) and G. Ellis (Galway) have computed the low-dimensiona integral homology of all Mathieu finite simple groups. [M.Dutour & G.Ellis, Wythoff polytopes and low-dimensional homology of Mathieu groups, J. Algebra 322 (2009), no. 11, 4143-4150.]

3. Mathieu Dutour (Marie Curie Fellow at Galway) and G. Ellis (Galway) have computed the low-dimensiona integral homology of infinite arithmetic groups such as PSL(4,Z) and PSL(3,Z[i]). This work is currently being written up.

4. Graham Ellis (Galway) and Paul Smith (Marie Curie Fellow at Galway) have used new techniques to recompute (and thus verify) the mod 2 cohomology rings of the groups of order 64. [G.Ellis & P.Smith, Computing group cohomology rings from the Lyndon-Hochschild-Serre spectral sequence. J. Symbolic Computation. In press, DOI:10.1016/j.jsc.2010.09.001] 4. Jie Wu (Singapore) and Roman Mikhailov (Moscow) have used the COHO software to obtain new results on homotopy groups of suspensions. ["On homotopy groups of the suspended classifying spaces", [J.Wu & R.Mikhailov, Algebraic and Geometric Topology 10 (2010), 565-625].

5. Bettina Eick, Heiko Dietrich and D¨orte Feichtenshlager ["Investigating p-groups by coclass with GAP", AMS Contemporary Mathematics 470 (2007), 45-63] have used the COHO software to investigate common cohomological properties of primepower groups of common coclass. 5. The COHO software has helped a number of other distinguished mathematicians with their research. For instance: a) It recently provided Alejandro Adem (British Columbia) with low-dimensional integral cohomology computations on symmetric groups of degree <=12; b) Marc Roeder (Marie Curie Fellow at Galway) provided Bartosz Putrycz (Gdansk/Leuven) with the integral cohomology of Hantzsche-Wendt (Bieberbach) groups of dimension <=8. c) Craig Westerland (Wisconsin) is using the COHO software to compute the cohomology of braid groups with certain twisted integral coefficients.

6. COHO software is being used by a PhD student at Galway to help with the automatic interpretation of X-Ray/Ultrasound images arising in the radiotherapy treatment of prostate cancer.