Turaev, Vladimir A norm for the cohomology of 2-complexes. (English) Zbl 1014.57002 Algebr. Geom. Topol. 2, 137-155 (2002). Summary: We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex \(X\), the norm on \(H^1(X;\mathbb R)\) determined by graphs on \(X\) majorates the Alexander-Fox norms derived from \(\pi_1(X)\). Reviewer: Andrei Vesnin (Novosibirsk) Cited in 9 Documents MSC: 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57M05 Fundamental group, presentations, free differential calculus Keywords:group cohomology; Alexander-Fox polynomials; norm; finite 2-complexes PDFBibTeX XMLCite \textit{V. Turaev}, Algebr. Geom. Topol. 2, 137--155 (2002; Zbl 1014.57002) Full Text: DOI arXiv EuDML EMIS References: [1] D Auckly, The Thurston norm and three-dimensional Seiberg-Witten theory, Osaka J. Math. 33 (1996) 737 · Zbl 0881.57034 [2] R H Fox, Free differential calculus II: The isomorphism problem of groups, Ann. of Math. \((2)\) 59 (1954) 196 · Zbl 0055.01704 · doi:10.2307/1969686 [3] P B Kronheimer, Minimal genus in \(S^1\times M^3\), Invent. Math. 135 (1999) 45 · Zbl 0917.57017 · doi:10.1007/s002220050279 [4] P B Kronheimer, T S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797 · Zbl 0851.57023 · doi:10.4310/MRL.1994.v1.n6.a14 [5] P B Kronheimer, T S Mrowka, Scalar curvature and the Thurston norm, Math. Res. Lett. 4 (1997) 931 · Zbl 0892.57011 · doi:10.4310/MRL.1997.v4.n6.a12 [6] C T McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. \((4)\) 35 (2002) 153 · Zbl 1009.57021 · doi:10.1016/S0012-9593(02)01086-8 [7] W P Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986) · Zbl 0585.57006 [8] V Turaev, Torsion invariants of \(\mathrm{Spin}^c\)-structures on 3-manifolds, Math. Res. Lett. 4 (1997) 679 · Zbl 0891.57019 · doi:10.4310/MRL.1997.v4.n5.a6 [9] V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag (2001) · Zbl 0970.57001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.