The Experts below are selected from a list of 1680 Experts worldwide ranked by ideXlab platform
Marco Spadini - One of the best experts on this subject based on the ideXlab platform.
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on the uniqueness of the fixed point index on differentiable manifolds
Fixed Point Theory and Applications, 2004Co-Authors: Massimo Furi, Maria Patrizia Pera, Marco SpadiniAbstract:It is well known that some of the properties enjoyed by the fixed point index can be chosen as axioms, the choice depending on the class of maps and spaces considered. In the context of finite-dimensional real differentiable manifolds, we will provide a simple proof that the fixed point index is uniquely determined by the properties of normalization, additivity, and Homotopy Invariance.
Goncalo Tabuada - One of the best experts on this subject based on the ideXlab platform.
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the gysin triangle via localization and a 1 Homotopy Invariance
Transactions of the American Mathematical Society, 2017Co-Authors: Goncalo Tabuada, Michel Van Den BerghAbstract:The first author was partially supported by the NSF CAREER Award #1350472 and by the Portuguese Foundation for Science and Technology grant PEst-OE/MAT/UI0297/2014.
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1 Homotopy Invariance of algebraic k theory with coefficients and du val singularities
Annals of K-Theory, 2017Co-Authors: Goncalo TabuadaAbstract:C. Weibel, and Thomason and Trobaugh, proved (under some assumptions) that algebraic K-theory with coefficients is A1-Homotopy invariant. We generalize this result from schemes to the broad setting of dg categories. Along the way, we extend the Bass–Quillen fundamental theorem as well as Stienstra’s foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-Homotopy Invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the du Val singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain vanishing and divisibility properties of algebraic K-theory (without coefficients).
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the gysin triangle via localization and a1 Homotopy Invariance
arXiv: Algebraic Geometry, 2015Co-Authors: Goncalo Tabuada, Michel Van Den BerghAbstract:Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-Homotopy invariant of dg categories E, we construct an associated Gysin triangle relating the value of E at the dg categories of perfect complexes of X, Z, and U. In the particular case where E is Homotopy K-theory, this Gysin triangle yields a new proof of Quillen's localization theorem, which avoids the use of devissage. As a first application, we prove that the value of E at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of E at the smooth projective schemes. As a second application, we compute the additive invariants of relative cellular spaces in terms of the bases of the corresponding cells. Finally, as a third application, we construct explicit bridges relating motivic Homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. This leads to a comparison between different motivic Gysin triangles as well as to an etale descent result concerning noncommutative mixed motives with rational coefficients.
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a1 Homotopy Invariance of algebraic k theory with coefficients and kleinian singularities
arXiv: K-Theory and Homology, 2015Co-Authors: Goncalo TabuadaAbstract:C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-Homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-Homotopy Invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-theory with coefficients of dg cluster categories using solely the kernel and cokernel of the Coxeter matrix. This leads to a complete computation of the algebraic K-theory with coefficients of the Kleinian singularities parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain some vanishing and divisibility properties of algebraic K-theory (without coefficients).
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the fundamental theorem via derived morita Invariance localization and 1 Homotopy Invariance
Journal of K-theory, 2012Co-Authors: Goncalo TabuadaAbstract:We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and 1-Homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in Homotopy algebraic K-theory, and periodic cyclic homology, respectively.
Varghese Mathai - One of the best experts on this subject based on the ideXlab platform.
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index type invariants for twisted signature complexes and Homotopy Invariance
Mathematical Proceedings of the Cambridge Philosophical Society, 2014Co-Authors: Moulay Tahar Benameur, Varghese MathaiAbstract:For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X, ${\cal E}$ ,H) for the twisted odd signature operator valued in a flat hermitian vector bundle ${\cal E}$ , where H = ∑ ij+1H2j+1 is an odd-degree closed differential form on X and H2j+1 is a real-valued differential form of degree 2j+1. We show that ρ(X, ${\cal E}$ ,H) is independent of the choice of metrics on X and ${\cal E}$ and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah–Patodi–Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The Homotopy Invariance of the rho invariant ρ(X, ${\cal E}$ ,H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.
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Index type invariants for twisted signature complexes and Homotopy Invariance
2013Co-Authors: Moulay Tahar Benameur, Varghese MathaiAbstract:For a closed, oriented, odd dimensional manifold X, we define the rho invariant rho(X,E,H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H = \sum i^{j+1} H_{2j+1} is an odd-degree closed differential form on X and H_{2j+1} is a real-valued differential form of degree {2j+1}. We show that the twisted rho invariant rho(X,E,H) is independent of the choice of metrics on X and E and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah-Patodi-Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The Homotopy Invariance of the rho invariant rho(X,E,H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.
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on the Homotopy Invariance of l 2 torsion for covering spaces
eprint arXiv:dg-ga 9706006, 1997Co-Authors: Varghese Mathai, Mel RothenbergAbstract:We prove the Homotopy Invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the Homotopy Invariance was established earlier by Lueck. We also give some applications of our results.
Massimo Furi - One of the best experts on this subject based on the ideXlab platform.
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on the uniqueness of the fixed point index on differentiable manifolds
Fixed Point Theory and Applications, 2004Co-Authors: Massimo Furi, Maria Patrizia Pera, Marco SpadiniAbstract:It is well known that some of the properties enjoyed by the fixed point index can be chosen as axioms, the choice depending on the class of maps and spaces considered. In the context of finite-dimensional real differentiable manifolds, we will provide a simple proof that the fixed point index is uniquely determined by the properties of normalization, additivity, and Homotopy Invariance.
G Skordev - One of the best experts on this subject based on the ideXlab platform.
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integer valued fixed point index for compositions of acyclic maps
Journal of Fixed Point Theory and Applications, 2007Co-Authors: E G Sklyarenko, G SkordevAbstract:An integer-valued fixed point index for compositions of acyclic multivalued maps is constructed. This index has the additivity, Homotopy Invariance, normalization, commutativity, and multiplicativity properties. The acyclicity is with respect to the Cech cohomology with integer coefficients. The technique of chain approximation is used.