5,969 research outputs found
The evaluation subgroup of a fibre inclusion
Given a fibration of simply connected CW complexes of finite type, we study
the evaluation subgroup of the fibre inclusion as an invariant of
fibre-homotopy type. For spherical fibrations, we show the evaluation subgroup
may be expressed as an extension of the Gottlieb group of the fibre sphere
provided the classifying map induces the trivial map on homotopy groups. We
extend this result after rationalization: We show that the rationalized
evaluation subgroup of the fibre inclusion decomposes as the direct sum of the
rationalized Gottlieb group of the fibre and the rationalized homotopy group of
the base if and only if the classifying map induces the trivial map on rational
homotopy groups.Comment: 15 page
Splitting off Rational Parts in Homotopy Types
It is known algebraically that any abelian group is a direct sum of a
divisible group and a reduced group (See Theorem 21.3 of
\cite{Fuchs:abelian-group}). In this paper, conditions to split off rational
parts in homotopy types from a given space are studied in terms of a variant of
Hurewicz map, say \bar{\rho} : [S_{\Q}^{n},X] \to H_n(X;\Z) and generalized
Gottlieb groups. This yields decomposition theorems on rational homotopy types
of Hopf spaces, -spaces and Gottlieb spaces, which has been known in various
situations, especially for spaces with finiteness conditions.Comment: 6 page
Interactive design exploration for constrained meshes
In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the design intent. In addition, they only offer limited support for exploring alternative feasible shapes, due to the high complexity of the optimization algorithm. We address these shortcomings and present a computational framework for interactive shape exploration of discrete geometric structures in the context of freeform architectural design. Our method is formulated as a mesh optimization subject to shape constraints. Our formulation can enforce soft constraints and hard constraints at the same time, and handles equality constraints and inequality constraints in a unified way. We propose a novel numerical solver that splits the optimization into a sequence of simple subproblems that can be solved efficiently and accurately. Based on this algorithm, we develop a system that allows the user to explore designs satisfying geometric constraints. Our system offers full control over the exploration process, by providing direct access to the specification of the design space. At the same time, the complexity of the underlying optimization is hidden from the user, who communicates with the system through intuitive interfaces
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