scholarly journals $G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture

2021 ◽  
Author(s):  
Yoshiyasu Fukumoto
Keyword(s):  
Homotopy Invariance ◽  
Novikov Conjecture

scholarly journals Topological rigidity problems

2016 ◽  
Vol 7 (4) ◽  
pp. 161
Author(s):  
Ramesh Kasilingam
Keyword(s):  
Group Ring ◽  
Whitehead Group ◽  
Homotopy Invariance ◽  
Open Problems ◽  
Novikov Conjecture ◽  
Additive Category ◽  
The Status ◽  
Borel Conjecture ◽  
Aspherical Manifolds ◽  
Rigidity Problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

scholarly journals A comment on the Novikov conjecture

1981 ◽  
Vol 83 (3) ◽  
pp. 656-656 ◽  
Author(s):  
Jerome Kaminker ◽  
John G. Miller
Keyword(s):  
Novikov Conjecture

scholarly journals Existence Theorems on Solvability of Constrained Inclusion Problems and Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Teffera M. Asfaw
Keyword(s):  
Reflexive Banach Space ◽  
Existence Theorems ◽  
Maximal Monotone ◽  
Existence Of Solution ◽  
Existence Results ◽  
Uniformly Convex ◽  
Homotopy Invariance ◽  
Inclusion Problems ◽  
Compact Resolvents ◽  
Nonlinear Variational Inequality

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎be a maximal monotone operator andC:X⊇D(C)→X⁎be bounded and continuous withD(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the typeT+Cprovided thatCis compact orTis of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition onT+C. The operatorCis neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.

scholarly journals Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

2017 ◽  
Vol 263 (11) ◽  
pp. 7162-7186 ◽  
Author(s):  
Marek Izydorek ◽  
Thomas O. Rot ◽  
Maciej Starostka ◽  
Marcin Styborski ◽  
Robert C.A.M. Vandervorst
Keyword(s):  
Hilbert Spaces ◽  
Conley Index ◽  
Homotopy Invariance ◽  
Morse Homology

scholarly journals The strong Suslin reciprocity law

2021 ◽  
Vol 157 (4) ◽  
pp. 649-676
Author(s):  
Daniil Rudenko
Keyword(s):  
Rational Functions ◽  
Projective Curve ◽  
Main Ingredient ◽  
Homotopy Invariance ◽  
Reciprocity Law ◽  
Hyperbolic Volume ◽  
Scissors Congruence ◽  
Dehn Invariant ◽  
Contracting Homotopy ◽  
Invariance Theorem

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$ -theory. The Milnor $K$ -groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

scholarly journals Stability and Hopf-Bifurcation Analysis of Delayed BAM Neural Network under Dynamic Thresholds

2009 ◽  
Vol 14 (4) ◽  
pp. 435-461 ◽  
Author(s):  
P. D. Gupta ◽  
N. C. Majee ◽  
A. B. Roy
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Global Bifurcation ◽  
Distributed Delay ◽  
Degree Theory ◽  
Homotopy Invariance ◽  
Normal Form Theory ◽  
Form Theory ◽  
Bam Neural Network ◽  
Uniqueness Of Equilibrium

In this paper the dynamics of a three neuron model with self-connection and distributed delay under dynamical threshold is investigated. With the help of topological degree theory and Homotopy invariance principle existence and uniqueness of equilibrium point are established. The conditions for which the Hopf-bifurcation occurs at the equilibrium are obtained for the weak kernel of the distributed delay. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and central manifold theorem. Lastly global bifurcation aspect of such periodic solutions is studied. Some numerical simulations for justifying the theoretical analysis are also presented.

scholarly journals The homotopy invariance of the Kuranishi space

1990 ◽  
Vol 34 (2) ◽  
pp. 337-367 ◽  
Author(s):  
William M. Goldman ◽  
John J. Millson
Keyword(s):  
Homotopy Invariance
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