Transactions of the American Mathematical Society

Author(s): Mihai Damian
Journal: Trans. Amer. Math. Soc. 361 (2009), 1791-1809.
MSC (2000): Primary 57R70, 55P15, 57Q10
Posted: October 30, 2008
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Abstract: Let

be a finite connected CW-complex of dimension

. If its fundamental group $\pi_1(X)$ is polycyclic of Hirsch number

, we show that at least one homotopy group $\pi_{i}(X)$ is not finitely generated. If

the same conclusion holds unless

is an Eilenberg-MacLane space $K(\pi_1(X),1)$ .

1.: L. Auslander, F. E. A. Johnson, On a conjecture of C.T.C. Wall, J. London Math. Soc. (2) 14 (1976), no. 2, 331-332. MR 0423362 (54:11341)
2.: M. Bestvina, N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445-470. MR 1465330 (98i:20039)
3.: R. Bieri, B. Eckmann, Finiteness properties of duality groups, Comment. Math. Helv. 49 (1974), 74-83. MR 0340450 (49:5205)
4.: R. Bieri, W. Neumann, R. Strebel, A geometric invariant for discrete groups, Invent. Math. 129 (1987), 451-477. MR 914846 (89b:20108)
5.: R. Bieri, B. Renz, Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv. 63 (1998), 464-497. MR 960770 (90a:20106)
6.: G. Bredon, Topology and Geometry, GTM 139, Springer Verlag, New York, 1997. MR 1700700 (2000b:55001)
7.: W. Browder, J. Levine, Fibering manifolds over the circle, Comment. Math. Helv. 40 (1966), 153-160. MR 0195104 (33:3309)
8.: K. S. Brown, Cohomology of groups, GTM 87, Springer Verlag, New York, 1982. MR 672956 (83k:20002)
9.: M. Damian, Formes fermées non singulières et propriétés de finitude des groupes Ann. Sci. Ec. Norm. Sup. 33 (2000) no. 3, 301-320. MR 1775183 (2002b:57031)
10.: M. Damian, On the higher homotopy groups of a finite CW-complex, Topology and its Appl. 149 (2005), 273-284. MR 2130870 (2005m:55013)
11.: F. T. Farrell, The obstruction to fibering a manifold over the circle, Bull. Amer. Math. Soc. 73 (1967), 737-740. MR 0215310 (35:6151)
12.: F. T. Farrell, W. C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. London Math. Soc. (2) 24 (1981), 308-324. MR 631942 (83b:20041)
13.: K. W Gruenberg, Cohomological Topics in Group Theory, L.N.M. 143, Springer Verlag, New York, 1970. MR 0279200 (43:4923)
14.: R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958. MR 0102797 (21:1583)
15.: K. A. Hirsch, On infinite soluble groups I, Proc. London Math. Soc. 44 (1938), 53-60.
16.: K. A. Hirsch, On infinite soluble groups IV, J. London Math. Soc. 27 (1952), 81-85. MR 0044526 (13:431g)
17.: G. Hochschild, J-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110-134. MR 0052438 (14:619b)
18.: B. Hu, Whitehead groups of finite polyhedra with non-positive curvature, Jour. of Diff. Geom. 38 3/1993, 501-517. MR 1243784 (94k:57037)
19.: F. Latour, Existence de 1-formes fermées non-singulières dans une classe de cohomologie de de Rham, Publ. Math. IHES 80 (1994). MR 1320607 (96f:57030)
20.: S. Maumary, Type simple d'homotopie, L.N.M. 48, Springer Verlag, New York, 1967.
21.: J. Meier, H. Meinert, L. Van Wyk, Finiteness properties and abelian quotients of graph groups, J. Math. Res. Lett. 3 (1996), 779-785. MR 1426535 (98b:57003)
22.: J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 0196736 (33:4922)
23.: S. P. Novikov, Multivalued functions and functionals. An analogue of the Morse theory, Soviet. Math. Dokl. vol 24, nr. 2/1981, 222-226. MR 630459 (83a:58025)
24.: A. Pajitnov, Surgery on the Novikov complex, K-theory 10 (1996), 323-412. MR 1404410 (97j:57043)
25.: A. Pajitnov Surgery on the Novikov complex, Rapport de Recherche, Nantes (1993), http://193.52.98.6/ pajitnov.
26.: A. Pajitnov, A. Ranicki, The Whitehead group of the Novikov ring, K-theory 21 (2000) no. 4, 325-365. MR 1828181 (2002g:19001)
27.: A. Ranicki, High-dimensional knot theory, Springer (1998). MR 1713074 (2000i:57044)
28.: A. Ranicki, Lower K- and L- theory, L.M.S. Lecture Notes, Cambridge, 1992. MR 1208729 (94f:19001)
29.: A. Ranicki, Finite domination and Novikov rings, Topology 34 (1995), 619-632. MR 1341811 (96j:57022)
30.: J. Stallings, The piecewise linear structure of the Euclidean space, Proc. Cambridge Phil. Soc. 58 (1962), 481-488. MR 0149457 (26:6945)
31.: J-P. Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math. (2) 58 (1953), 258-294. MR 0059548 (15:548c)
32.: L. Siebenmann, A total Whitehead obstruction to fibering over the circle, Comment. Math. Helv. 45 (1970), 1-48. MR 0287564 (44:4768)
33.: J-C. Sikorav, Homologie de Novikov associée à une classe de cohomologie réelle de dégré un, Thèse Orsay, 1987.
34.: E. Spanier, Algebraic Topology, Mc Graw-Hill, New York, 1966. MR 0210112 (35:1007)
35.: D. Tischler, On fibering certain foliated manifolds over $S^{1}$ , Topology 9 (1970), 153-154. MR 0256413 (41:1069)
36.: C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), 56-69. MR 0171284 (30:1515)
37.: C. T. C. Wall, Surgery on compact manifolds, London Math. Society Monographs, No. 1. Academic Press, London-New York, 1970. MR 0431216 (55:4217)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R70, 55P15, 57Q10

Mihai Damian
Affiliation: Irma, Université Louis Pasteur, 7, rue René Descartes, 67 084 Strasbourg, France
Email: damian@math.u-strasbg.fr

DOI: 10.1090/S0002-9947-08-04632-1
PII: S 0002-9947(08)04632-1
Keywords: Novikov homology, closed $1$-forms, CW-complexes, homotopy groups
Received by editor(s): January 26, 2007
Posted: October 30, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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