Investigations in the homotopy theory of continuous mappings. I. The algebraic theory of systems. II. The natural system and homotopy type

Postnikov, M. M.

doi:10.1090/trans2/007/01

Cited by 8 publications

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Unfortunately, both Postnikov's encoding of the homotopy type and his algebraic reconstruction of the cohomology are rather complicated, do not appear to have gained much popularity. They seem to be fully described only in his original monograph [66] or its translation [67], both being rather obscure references. At the moment, it is not clear to us what is the modern state of the art in terms of reconstructing the cohomology of a space with coefficients in a local system in terms of the space's homotopy type.…”

Section: Cohomology With Local Coefficientsmentioning

confidence: 99%

The Calabi complex and Killing sheaf cohomology

Khavkine

2017

Journal of Geometry and Physics

View full text Add to dashboard Cite

It has recently been noticed that the degeneracies of the Poisson bracket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of differential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of locally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of generalized Poincar\'e duality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds.Comment: tikz-cd diagrams, 69 page

show abstract

Section: Cohomology With Local Coefficientsmentioning

confidence: 99%

The Calabi complex and Killing sheaf cohomology

Khavkine

2017

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…Remark 2.5 Given a topological space X consider the exterior space X tr which has the trivial externology ε = {X} and the exterior space X to which has the total externology. If X [n] denotes the Postnikov section in standard homotopy theory, see [15] , [16] , then the standard sections (X [n] ) tr are weak B-equivalent to the new sections P B n (X tr ) . Nevertheless, the sections P B n (X to ) are trivial in Ho(E) .…”

Section: Postnikov Factorization For Brown-grossman Groupsmentioning

confidence: 99%

Postnikov factorizations at infinity

Aldana

Paricio

Rodríguez

2005

Topology and its Applications

View full text Add to dashboard Cite

“…Postnikov proved this for the case B is a point [54]; Moore then proved the general case, in the setting of semisimplicial complexes [46]. Hermann [20] gave the proof for an ordinary fibration, as did Eckmann-Hilton (see [22]).…”

Section: (Postnikov J Moore) There Exists a Postnikov Resolution Fmentioning

confidence: 99%

“…In recent years a different approach to obstruction theory has evolved, initiated by the work of Postnikov [54]. There are two important aspects to this more recent approach:…”

mentioning

confidence: 99%