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We consider an effective action of a compact pn´1q-torus on a smooth 2nmanifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n´1 has a specific topology which we axiomatize in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points.2010 Mathematics Subject Classification. Primary 55R25, 57N65 ; Secondary 55R40, 55R55, 55R91, 57N40, 57N80, 57S15.

For a simplicial complex K on m vertices and simplicial complexes K 1 , . . . , K m , we introduce a new simplicial complex K(K 1 , . . . , K m ), called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari, M. Bendersky, F. R. Cohen, and S. Gitler. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes P (P 1 , . . . , P m ), as introduced by G. Agnarsson. The substitution gives rise to an operad structure on the set of finite simplicial complexes in which a simplicial complex on m vertices is considered as an m-ary operation. We prove the following main results: (1) the complex K(K 1 , . . . , K m ) is a simplicial sphere if and only if K is a simplicial sphere and the K i are the boundaries of simplices, (2) the class of spherical nerve-complexes is closed under substitution, (3) multigraded betti numbers of K(K 1 , . . . , K m ) are expressed in terms of those of the original complexes K, K 1 , . . . , K m . We also describe connections between the obtained results and the known results of other authors.2010 Mathematics Subject Classification. Primary 05E45; Secondary 52B11, 52B05, 55U10, 13F55. download from IP 128.122.253.212.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 176 A. A. AYZENBERG question is how to describe the properties of the Stanley-Reisner algebra k[K P ] and the cohomology ring H * (Z P ; k) ∼ = Tor * , * k[m] (k[K P ], k) for arbitrary convex polytopes. In the work of A. Bahri, M. Bendersky, F. R Cohen, and S. Gitler [5] one finds a construction that allows, for a given simple polytope P with m facets and a set of natural numbers (l 1 , . . . , l m ), the building of a new simple polytope P (l 1 , . . . , l m ). The simplicial complex ∂P (l 1 , . . . , l m ) * can be described combinatorially in terms of minimal non-simplices. This approach allows us to describe the moment-angle complex Z ∂P (l 1 ,...,l m ) * (D 2 , S 1 ) as the polyhedral product Z P * ((D 2l i , S 2l i −1 )), which yields a more efficient description of the cohomology ring H * (Z P (l 1 ,...,l m ) ), as well as of the cohomology rings of some quasitoric manifolds over such polytopes.The use of arbitrary (i.e., not necessarily simple) polytopes in toric topology yields a wider class of examples and more general constructions. One such construction is known in convex geometry (see, for example, [1]) and, for a given a polytope P ⊂ R m and polytopes P 1 , . . . , P m , produces a new polytope P (P 1 , . . . , P m ). In general, the combinatorial structure of P (P 1 , . . . , P m ) depends on the chosen geometric representation of P ⊂ R m . However, under some conditions, the construction can be made combinatorial, i.e., we may assume that the face poset of P (P 1 , . . . , P m ) depends only on the face posets of the original polytopes P, P 1 , . . . , P m . In the particular case when P i = l i is a simplex on l i vertices, the polytope P (P 1 , . . . , P m ) c...

Abstract. We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan ∆ determines a volume polynomial V ∆ whose values are the volumes of multi-polytopes based on ∆. This homogeneous polynomial is further used to construct a Poincare duality algebra A˚p∆q. We study the structure and properties of V ∆ and A˚p∆q and give applications and connections to other subjects, such as Macaulay duality, Novik-Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the g-theorem does not hold for multi-polytopes.