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Two remarks on Wall's D2 problem



If a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X) = G is finite, and χ(X) ≥ 1 − def(G), then XS2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on G and χ(X).



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Research partially supported by NSERC Discovery Grant A4000.



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[1] Bass, H. Algebraic K-theory. (Benjamin, W. A., Inc., New York-Amsterdam, 1968).
[2] Bestvina, M. and Brady, N. Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), 445470.
[3] Bridson, M. R. and Tweedale, M. Deficiency and abelianised deficiency of some virtually free groups. Math. Proc. Camb. Philos. Soc. 143 (2007), 257264.
[4] Brown, K. A. Relation modules of polycyclic-by-finite groups. J. Pure Appl. Algebra 20 (1981), 227239.
[5] Brown, K. S. and Kahn, P. J. Homotopy dimension and simple cohomological dimension of spaces. Comment. Math. Helv. 52 (1977), 111127.
[6] Browning, W. J. Homotopy types of certain finite cw-complexes with finite fundametal group. (ProQuest LLC, Ann Arbor, MI, 1978) Ph.D. thesis. Cornell University.
[7] Cohen, J. M. Complexes of cohomological dimension two. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII (Amer. Math. Soc., Providence, R.I., 1978), pp. 221223.
[8] Coxeter, H. S. M. and Moser, W. O. J. Generators and relations for discrete groups. fourth ed. Ergeb. Math. Grenzgeb. [Results in Mathematics and Related Areas], vol. 14 (Springer-Verlag, Berlin-New York, 1980).
[9] Dunwoody, M. J. The homotopy type of a two-dimensional complex. Bull. London Math. Soc. 8 (1976), 282285.
[10] Eilenberg, S. and Ganea, T. On the Lusternik–Schnirelmann category of abstract groups. Ann. of Math. (2) 65 (1957), 517518.
[11] Hambleton, I. and Kreck, M. Cancellation of lattices and finite two-complexes. J. Reine Angew. Math. 442 (1993), 91109.
[12] Hambleton, I., Pamuk, S. and Yalçin, E. Equivariant CW-complexes and the orbit category. Comment. Math. Helv. 88 (2013), 369425.
[13] Harlander, J. Some aspects of efficiency. Groups—Korea '98 (Pusan) (de Gruyter, Berlin, 2000) pp. 165–180.
[14] Harlander, J. On the relation gap and relation lifting problem. Groups St Andrews 2013 London Math. Soc. Lecture Note Ser. vol. 422 (Cambridge University Press, Cambridge, 2015), pp. 278–285.
[15] Harlander, J. and Jensen, J. A. Exotic relation modules and homotopy types for certain 1-relator groups. Algebr. Geom. Topol. 6 (2006), 21632173.
[16] Harlander, J. and Jensen, J. A. On the homotopy type of CW-complexes with aspherical fundamental group. Topology Appl. 153 (2006), 30003006.
[17] Harlander, J. and Misseldine, A. On the K-theory and homotopy theory of the Klein bottle group. Homology Homotopy Appl. 13 (2011), 6372.
[18] Hog-Angeloni, C. and Metzler, W. (eds.). Two-dimensional homotopy and combinatorial group theory. London Mathematical Society Lecture Note Series, vol. 197 (Cambridge University Press, Cambridge, 1993).
[19] Ji, F. and Ye, S. Partial Euler characteristics, normal generations and the stable D(2) problem. arXiv:1503.01987 (2015).
[20] Jin, X., Su, Y. and Yu, L. Homology roses and the D(2)-problem. Sci. China Math. 58 (2015), 17531770.
[21] Johnson, F. E. A. Stable modules and the D(2)-problem. London Math. Soc. Lecture Note Series, vol. 301 (Cambridge University Press, Cambridge, 2003).
[22] Magurn, B. A., van der Kallen, W. and Vaserstein, L. N. Absolute stable rank and Witt cancellation for noncommutative rings. Invent. Math. 91 (1988), 525542.
[23] Mannan, W. H. The D(2) property for D 8. Algebr. Geom. Topol. 7 (2007), 517528.
[24] Mannan, W. H. Quillen's plus construction and the D(2) problem. Algebr. Geom. Topol. 9 (2009), 13991411.
[25] Mannan, W. H. Realising algebraic 2-complexes by cell complexes. Math. Proc. Camb. Phil. Soc. 146 (2009), 671673.
[26] Mannan, W. H. and O'Shea, S. Minimal algebraic complexes over D 4n. Algebr. Geom. Topol. 13 (2013), 32873304.
[27] McConnell, J. C. and Robson, J. C. Noncommutative Noetherian rings. revised ed. Graduate Studies in Math., vol. 30 (American Mathematical Society, Providence, RI, 2001), With the cooperation of Small, L. W.
[28] O'Shea, S. The D(2)-problem for dihedral groups of order 4n. Algebr. Geom. Topol. 12 (2012), 22872297.
[29] Stafford, J. T. Stable structure of noncommutative Noetherian rings. J. Algebra 47 (1977), 244267.
[30] Stafford, J. T. Absolute stable rank and quadratic forms over noncommutative rings. K-Theory 4 (1990), 121130.
[31] Swan, R. G. Minimal resolutions for finite groups. Topology 4 (1965), 193208.
[32] Wall, C. T. C. Finiteness conditions for CW-complexes. Ann. of Math. (2) 81 (1965), 5669.
[33] Wall, C. T. C. Finiteness conditions for cw complexes. II. Proc. Roy. Soc. Ser. A 295 (1966), 129139.
[34] Whitehead, J. H. C. Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 45 (1939), 243327.

Two remarks on Wall's D2 problem



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