In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.
Source Osaka J. Zentralblatt MATH identifier Subjects Primary: 57R Equivariant algebraic topology of manifolds.
Masuda, Mikiya; Panov, Taras. On the cohomology of torus manifolds. Osaka J.
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Abstract Article info and citation First page References Abstract A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. Article information Source Osaka J. Export citation. This established link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits.
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Conversely, subtle properties of a combinatorial object can be realized by interpreting it as the orbit structure for a proper manifold or as a complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics.
The exposition centers around the theory of moment-angle complexes, providing an effective way to study invariants of triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook.
It will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved. Graduate students and research mathematicians interested in topology or combinatorics; topologists interested in combinatorial applications and vice versa.
The text contains a wealth of material and … the book may be a welcome collection for researchers in the field and a useful overview of the literature for novices. AMS Homepage.
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Ordering on the AMS Bookstore is limited to individuals for personal use only. The book is quite well-written and includes many new and well-known open problems -- Mathematical Reviews The text contains a wealth of material and … the book may be a welcome collection for researchers in the field and a useful overview of the literature for novices. Advanced search. Author s Product display : Victor M.
Buchstaber ; Taras E.
Masuda , Panov : On the cohomology of torus manifolds
Abstract: The book presents the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. Volume: Publication Month and Year: Copyright Year: Page Count: Cover Type: Softcover. Print ISBN Online ISBN Print ISSN: