Geodesic
Minimal algebraic complexes over D4n
Mannan, Wajid H1 ; O’Shea, Seamus2
1Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
2Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3287-3304
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We show that cancellation of free modules holds in the stable class Ω3(ℤ) over dihedral groups of order 4n. In light of a recent result on realizing k–invariants for these groups, this completes the proof that all dihedral groups satisfy the D(2) property.

DOI : 10.2140/agt.2013.13.3287
Classification : 57M20, 16E05, 16E10, 55P15, 55Q20
Keywords: non-simply connected homotopy, cancellation of modules, Wall's D(2) problem, algebraic homotopy

Mannan, Wajid H  1   ; O’Shea, Seamus  2

1 Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
2 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 
@article{10_2140_agt_2013_13_3287,
     author = {Mannan, Wajid H and O{\textquoteright}Shea, Seamus},
     title = {Minimal algebraic complexes over {D4n}},
     journal = {Algebraic and Geometric Topology},
     pages = {3287--3304},
     year = {2013},
     volume = {13},
     number = {6},
     doi = {10.2140/agt.2013.13.3287},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.3287/}
}
TY  - JOUR
AU  - Mannan, Wajid H
AU  - O’Shea, Seamus
TI  - Minimal algebraic complexes over D4n
JO  - Algebraic and Geometric Topology
PY  - 2013
SP  - 3287
EP  - 3304
VL  - 13
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.3287/
DO  - 10.2140/agt.2013.13.3287
ID  - 10_2140_agt_2013_13_3287
ER  - 
%0 Journal Article
%A Mannan, Wajid H
%A O’Shea, Seamus
%T Minimal algebraic complexes over D4n
%J Algebraic and Geometric Topology
%D 2013
%P 3287-3304
%V 13
%N 6
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.3287/
%R 10.2140/agt.2013.13.3287
%F 10_2140_agt_2013_13_3287
Mannan, Wajid H; O’Shea, Seamus. Minimal algebraic complexes over D4n. Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3287-3304. doi: 10.2140/agt.2013.13.3287

[1] M R Bridson, M Tweedale, Deficiency and Abelianized deficiency of some virtually free groups, Math. Proc. Cambridge Philos. Soc. 143 (2007) 257

[2] T Edwards, Generalised Swan modules and the $\mathrm{D(2)}$ problem, Algebr. Geom. Topol. 6 (2006) 71

[3] I Hambleton, M Kreck, Cancellation of lattices and finite two-complexes, J. Reine Angew. Math. 442 (1993) 91

[4] F E A Johnson, Stable modules and the structure of Poincaré $3$–complexes, from: "Geometry and topology" (editors K Grove, I H Madsen, E K Pedersen), Contemp. Math. 258, Amer. Math. Soc. (2000) 227

[5] F E A Johnson, Explicit homotopy equivalences in dimension two, Math. Proc. Cambridge Philos. Soc. 133 (2002) 411

[6] F E A Johnson, Stable modules and the $\mathrm{D}(2)$–problem, London Mathematical Society Lecture Note Series 301, Cambridge Univ. Press (2003)

[7] F E A Johnson, Minimal $2$–complexes and the $\mathrm D(2)$–problem, Proc. Amer. Math. Soc. 132 (2004) 579

[8] M P Latiolais, When homology equivalence implies homotopy equivalence for $2$–complexes, J. Pure Appl. Algebra 76 (1991) 155

[9] W H Mannan, Homotopy types of truncated projective resolutions, Homology, Homotopy Appl. 9 (2007) 445

[10] W H Mannan, The $\mathrm{D}(2)$ property for $D_8$, Algebr. Geom. Topol. 7 (2007) 517

[11] W H Mannan, Periodic cohomology, Homology, Homotopy Appl. 10 (2008) 135

[12] W H Mannan, Quillen's plus construction and the $\mathrm{D}(2)$ problem, Algebr. Geom. Topol. 9 (2009) 1399

[13] W H Mannan, Realizing algebraic $2$–complexes by cell complexes, Math. Proc. Cambridge Philos. Soc. 146 (2009) 671

[14] S O’Shea, The $\mathrm{D}(2)$–problem for dihedral groups of order $4n$, Algebr. Geom. Topol. 12 (2012) 2287

[15] J R Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. 88 (1968) 312

[16] R G Swan, Groups of cohomological dimension one, J. Algebra 12 (1969) 585

[17] R G Swan, Torsion free cancellation over orders, Illinois J. Math. 32 (1988) 329

[18] C T C Wall, Finiteness conditions for $\mathrm{CW}$–complexes, Ann. of Math. 81 (1965) 56

Cité par Sources :