## Geometry & Topology

### Intersections of quadrics, moment-angle manifolds and connected sums

#### Abstract

For the intersections of real quadrics in $ℝn$ and in $ℂn$ associated to simple polytopes (also known as universal abelian covers and moment-angle manifolds, respectively) we obtain the following results:

(1)  Every such manifold of dimension greater than or equal to 5, connected up to the middle dimension and with free homology, is diffeomorphic to a connected sum of sphere products. The same is true for the manifolds in infinite families stemming from each of them. This includes the moment-angle manifolds for which the result was conjectured by F Bosio and L Meersseman.

(2)  The topological effect on the manifolds of cutting off vertices and edges from the polytope is described. Combined with the result in (1), this gives the same result for many more natural, infinite families.

(3)  As a consequence of (2), the cohomology rings of the two manifolds associated to a polytope need not be isomorphic, contradicting published results about complements of arrangements.

(4)  Auxiliary but general constructions and results in geometric topology.

#### Article information

Source
Geom. Topol., Volume 17, Number 3 (2013), 1497-1534.

Dates
Revised: 11 December 2012
Accepted: 14 January 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732612

Digital Object Identifier
doi:10.2140/gt.2013.17.1497

Mathematical Reviews number (MathSciNet)
MR3073929

Zentralblatt MATH identifier
1276.14087

Keywords

#### Citation

Gitler, Samuel; López de Medrano, Santiago. Intersections of quadrics, moment-angle manifolds and connected sums. Geom. Topol. 17 (2013), no. 3, 1497--1534. doi:10.2140/gt.2013.17.1497. https://projecteuclid.org/euclid.gt/1513732612

#### References

• D Allen, J La Luz, A counterexample to a conjecture of Bosio and Meersseman, from: “Toric topology”, (M Harada, Y Karshon, M Masuda, T Panov, editors), Contemp. Math. 460, Amer. Math. Soc. (2008) 37–45
• A Bahri, M Bendersky, F R Cohen, S Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Adv. Math. 225 (2010) 1634–1668
• A Bahri, M Bendersky, F R Cohen, S Gitler, Cup-products for the polyhedral product functor, Math. Proc. Cambridge Philos. Soc. 153 (2012) 457–469
• A Bahri, M Bendersky, F R Cohen, S Gitler, Operations on polyhedral products and a new topological construction of infinite families of toric manifolds (2012)
• I V Baskakov, Cohomology of $K$–powers of spaces and the combinatorics of simplicial divisions, Uspekhi Mat. Nauk 57 (2002) 147–148
• I V Baskakov, Triple Massey products in the cohomology of moment-angle complexes, Uspekhi Mat. Nauk 58 (2003) 199–200 In Russian; translated in Russian Math. Surveys 58:5 (2003) 1039–1041
• F Bosio, Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano–Verjovsky, Ann. Inst. Fourier (Grenoble) 51 (2001) 1259–1297
• F Bosio, L Meersseman, Real quadrics in $\mathbb C\sp n$, complex manifolds and convex polytopes, Acta Math. 197 (2006) 53–127
• W Browder, Surgery on simply-connected manifolds, Ergeb. Math. Grenzgeb. 65, Springer, New York (1972)
• V M Buchstaber, T E Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series 24, Amer. Math. Soc. (2002)
• C Camacho, N H Kuiper, J Palis, The topology of holomorphic flows with singularity, Inst. Hautes Études Sci. Publ. Math. (1978) 5–38
• M Chaperon, Géométrie différentielle et singularités de systèmes dynamiques, Astérisque 138–139, Société Mathématique de France, Paris (1986)
• M Chaperon, S López De Medrano, Birth of attracting compact invariant submanifolds diffeomorphic to moment-angle manifolds in generic families of dynamics, C. R. Math. Acad. Sci. Paris 346 (2008) 1099–1102
• M W Davis, T Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991) 417–451
• G Denham, A I Suciu, Moment-angle complexes, monomial ideals and Massey products, Pure Appl. Math. Q. 3 (2007) 25–60 Special Issue: In honor of Robert D. MacPherson, Part 3
• V Gasharov, I Peeva, V Welker, Coordinate subspace arrangements and monomial ideals, from: “Computational commutative algebra and combinatorics”, (T Hibi, editor), Adv. Stud. Pure Math. 33, Math. Soc. Japan, Tokyo (2002) 65–74
• F González Acuña, Open Books, Lecture Notes, University of Iowa
• B Grünbaum, Convex polytopes, 2nd edition, Graduate Texts in Mathematics 221, Springer, New York (2003)
• A Haefliger, Differentiable imbeddings, Bull. Amer. Math. Soc. 67 (1961) 109–112
• F Hirzebruch, private conversation (1986)
• M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504–537
• J J Loeb, M Nicolau, On the complex geometry of a class of non-Kählerian manifolds, Israel J. Math. 110 (1999) 371–379
• M de Longueville, The ring structure on the cohomology of coordinate subspace arrangements, Math. Z. 233 (2000) 553–577
• D McGavran, Adjacent connected sums and torus actions, Trans. Amer. Math. Soc. 251 (1979) 235–254
• S López de Medrano, The space of Siegel leaves of a holomorphic vector field, from: “Holomorphic dynamics”, (X Gómez-Mont, J Seade, A Verjovsky, editors), Lecture Notes in Math. 1345, Springer, Berlin (1988) 233–245
• S López de Medrano, Topology of the intersection of quadrics in $\mathbb R\sp n$, from: “Algebraic topology”, (G Carlsson, R L Cohen, H R Miller, D C Ravenel, editors), Lecture Notes in Math. 1370, Springer, Berlin (1989) 280–292
• S López de Medrano, A Verjovsky, A new family of complex, compact, nonsymplectic manifolds, Bol. Soc. Brasil. Mat. 28 (1997) 253–269
• L Meersseman, A new geometric construction of compact complex manifolds in any dimension, Math. Ann. 317 (2000) 79–115
• L Meersseman, A Verjovsky, Holomorphic principal bundles over projective toric varieties, J. Reine Angew. Math. 572 (2004) 57–96
• L Meersseman, A Verjovsky, Sur les variétés LV–M, from: “Singularities II”, (J-P Brasselet, J L Cisneros-Molina, D Massey, J Seade, B Teissier, editors), Contemp. Math. 475, Amer. Math. Soc. (2008) 111–134
• C T C Wall, Classification of $(n-1)$–connected $2n$–manifolds, Ann. of Math. 75 (1962) 163–189
• C T C Wall, Classification problems in differential topology VI, Classification of $(s-1)$–connected $(2s+1)$–manifolds, Topology 6 (1967) 273–296
• C T C Wall, Stability, pencils and polytopes, Bull. London Math. Soc. 12 (1980) 401–421