Generalized manifolds, normal invariants, and 𝕃-homology

Let $X^{n}$ be an oriented closed generalized $n$ -manifold, $n\ge 5$ . In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$ -manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$ . An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$ -manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

On Steenrod 𝕃-homology, generalized manifolds, and surgery

The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.

Generalized almost para-contact manifolds

In this paper, we study generalized almost para-contact manifolds and obtain normality conditions in terms of classical tensor fields. We show that such manifolds naturally carry certain Lie bialgebroid/quasi-Lie algebroid structures on them and we relate these new generalized manifolds with classical almost para-contact manifolds. The paper contains several examples and a short review for relations between generalized geometry and string theory.

A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints

This paper presents a single, unified, and efficient algorithm for animating the coupler motions of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation for each type of four-bar linkages separately. Recently, we developed a unified algorithm for synthesizing various four-bar linkages by mapping planar displacements from Cartesian space to the image space using planar quaternions. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of generalized manifolds (or G-manifolds) that best fit the image points in the least squares sense. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple's iOS platform.