## Algebraic & Geometric Topology

### Homotopy groups and twisted homology of arrangements

Richard Randell

#### Abstract

Recent work of M Yoshinaga [Topology Appl. 155 (2008) 1022-1026] shows that in some instances certain higher homotopy groups of arrangements map onto nonresonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generalizing both results.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1299-1308.

Dates
Revised: 5 May 2009
Accepted: 6 May 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513797030

Digital Object Identifier
doi:10.2140/agt.2009.9.1299

Mathematical Reviews number (MathSciNet)
MR2520401

Zentralblatt MATH identifier
1173.55003

#### Citation

Randell, Richard. Homotopy groups and twisted homology of arrangements. Algebr. Geom. Topol. 9 (2009), no. 3, 1299--1308. doi:10.2140/agt.2009.9.1299. https://projecteuclid.org/euclid.agt/1513797030

#### References

• D C Cohen, Triples of arrangements and local systems, Proc. Amer. Math. Soc. 130 (2002) 3025–3031
• D C Cohen, A Dimca, P Orlik, Nonresonance conditions for arrangements, Ann. Inst. Fourier $($Grenoble$)$ 53 (2003) 1883–1896
• D C Cohen, A I Suciu, On Milnor fibrations of arrangements, J. London Math. Soc. $(2)$ 51 (1995) 105–119
• A Dimca, Ş Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. $(2)$ 158 (2003) 473–507
• A Dimca, Ş Papadima, Equivariant chain complexes, twisted homology and relative minimality of arrangements, Ann. Sci. École Norm. Sup. $(4)$ 37 (2004) 449–467
• A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
• A Hattori, Topology of $C\sp{n}$ minus a finite number of affine hyperplanes in general position, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975) 205–219
• R Liu, On the classification of $(n-k+1)$–connected embeddings of $n$–manifolds into $(n+k)$-manifolds in the metastable range, Trans. Amer. Math. Soc. 347 (1995) 4245–4258
• P Orlik, H Terao, Arrangements of hyperplanes, Grund. der Math. Wissenschaften 300, Springer, Berlin (1992)
• R Randell, Homotopy and group cohomology of arrangements, Topology Appl. 78 (1997) 201–213
• R Randell, Morse theory, Milnor fibers and minimality of hyperplane arrangements, Proc. Amer. Math. Soc. 130 (2002) 2737–2743
• M Yoshinaga, Generic section of a hyperplane arrangement and twisted Hurewicz maps, Topology Appl. 155 (2008) 1022–1026