By Victor M. Buchstaber and Taras E. Panov

The booklet offers the learn of torus activities on topological areas is gifted as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This demonstrated hyperlink is helping in figuring out the geometry and topology of an area with torus motion by way of learning the combinatorics of the gap of orbits. Conversely, sophisticated houses of a combinatorial item will be learned by way of studying it because the orbit constitution for a formal manifold or as a fancy acted on via a torus. The latter could be a symplectic manifold with Hamiltonian torus motion, a toric kind or manifold, a subspace association supplement, etc., whereas the combinatorial items comprise simplicial and cubical complexes, polytopes, and preparations. This technique additionally presents a typical topological interpretation by way of torus activities of many buildings from commutative and homological algebra utilized in combinatorics. The exposition facilities round the conception of moment-angle complexes, supplying a great way to review invariants of triangulations by means of equipment of equivariant topology. The ebook contains many new and famous open difficulties and will be appropriate as a textbook. it is going to be priceless for experts either in topology and in combinatorics and should aid to set up even tighter connections among the topics concerned

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And i < dim(link ), He i (link ; k) = 0. 18. A simplicial sphere is a Cohen{Macaulay complex. 16 shows that the h-vector of a simplicial sphere is an M -vector. 33) to simplicial spheres. 19 (Upper Bound Theorem for spheres, Stanley 123]). The hvector (h0 ; h1 ; : : : ; hn ) of a simplicial (n ? 1 with m vertices satis es ? 1 ; 0 6 i < n2 : Hence, the UBC holds for simplicial spheres, that is, ? 1 ) 6 fi C n (m) for i = 2; : : : ; n ? 35). A0 Proof. 32). In particular, dimk A2 = h1 = m ? n. Since A is generated by A2 , the number hi cannot exceed the?

1 to be a free k v1 ; : : : ; vm ]module with k1 generators in the corresponding degrees, and so on. i+1 ! i to be a free module with the corresponding generators. The Hilbert syzygy theorem guarantees this process to end up at most at the m-th step. 3. 23 (minimal resolution). For graded nitely generated modules M a minimal generator set (or a minimal basis ) can be chosen. This is done as follows. Take the lowest degree in which M is non-zero and there choose a vector space basis. Span a module M1 by this basis and then take the lowest degree in which M 6= M1 .

Note that the ideal IK is monomial and has basis of monomials v corresponding to missing faces of K . 3. Every square-free monomial ideal in the polynomial ring has the form IK for some simplicial complex K . Proof. Let I be a square-free monomial ideal. Set K=f m] : v 2= Ig: Then one easily checks that K is a simplicial complex and I = IK . 4. Let : K1 ! 7) between two simplicial complexes K1 and K2 on the vertex sets m1 ] and m2 ] respectively. De ne the map : k w1 ; : : : ; wm2 ] ! 1 fjg vi : Then descends to a homomorphism k(K2 ) !

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