Kevin P. Knudson
Let E be an elliptic curve defined via a Weierstrass equation F(x,y)=0 over an infinite field k. Denote by A the coordinate ring of E. In this note we compute the integral homology of PGL_2(A). We obtain a rigidity result as a corollary.
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We determine the integral homology of PSL4(Z) in degrees at most 5 and determine its p-part in higher degrees for the primes p>=5. Our method applies to other arithmetic groups; as illustrations we include descriptions of the integral homology of PGL3(Z[i]) and PGL3(Z[\exp(2π{i}/3)]) in degrees at most 5.
The low dimensional homology of projective linear group of rank two
本文研究了交换环上二阶射影线性群的低维同调，证明了局部整环上的 Bloch-Wigner 型正合序列，并导出了其二阶和三阶同调群与 K 理论的同构关系。
On the homology of special unitary groups over polynomial rings
本文研究P^1上最小非等平凡群概形SU_{3,P^1}的同伦不变性，证明存在自然同态诱导PGL_2(F)与G(F[t])的整同调同构，并借助算术子群基本域分析G(F[t,t^{-1}])的有理同调。
In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket.
The note provides a description of the homology of $GL_3$ over function rings of affine elliptic curves over arbitrary fields, following the earlier work of Takahashi and Knudson in the case $GL_2$. Some prospects for applications to K-theory of elliptic curves are also discussed.
We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.
We establish the stable homotopy classification of elliptic pseudodifferential operators on manifolds with corners and show that the set of elliptic operators modulo stable homotopy is isomorphic to the K-homology group of some stratified manifold. By way of application, generalizations of some recent results due to Monthubert and Nistor are given.
Fourier--Mukai equivalences for formal groups and elliptic Hochschild homology
本文比较两种椭圆 Hochschild 同调定义，提出形式群的新 Fourier--Mukai 对偶性以统一扭曲 Hochschild 同调，并引入全局版本暗示了普遍的 TMF-Hochschild 同调形式。
If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of square integrable sections of $S$ with the canonical left $C(M)$-action and the operator $χ(D)$ for $χ$ a normalizing function is a Fredholm module, and its $K$-homology class is independent of $χ$. In this expository article, we provide a detailed proof of this fact following the outline in the book "Analytic K-homology" by Higson and Roe.
It is well known that elliptic operators on a smooth compact manifold are classified by K-homology. We prove that a similar classification is also valid for manifolds with simplest singularities: isolated conical points and fibered boundary. The main ingredients of the proof of these results are: an analog of the Atiyah-Singer difference construction in the noncommutative case and an analog of Poincare isomorphism in K-theory for our singular manifolds. As applications we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with singularities and a formula …
Uniformly elliptic boundary value problems
本文通过统一K-同调理论研究非紧流形上均匀椭圆算子的边界条件，建立了与边界上的统一更高ρ-不变量的联系，并证明了去局部化APS索引定理。
We obtain a classification of elliptic operators modulo stable homotopy on manifolds with edges (this is in some sense the simplest class of manifolds with nonisolated singularities). We show that the operators are classified by the K-homology group of the manifold. To this end, we construct exact sequence in elliptic theory isomorphic to the K-homology exact sequence. The main part of the proof is the computation of the boundary map using semiclassical quantization.
Regular boundary conditions and analytic K-homology
本文推广Baum-Douglas-Taylor的结果，利用正则边界条件的系统方法，将局部边界条件定义的K-同调循环推广到任意阶椭圆算子及非紧致流形，并证明在更强正则性下可定义绝对K-同调类。
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our framework ideas coming from Baum-Douglas geometric $K$-homology and in particular we introduce a notion of geometric cycles, that can be categorized as a variant of the famous geometric $K$-homology groups, for the specific situation here. We also define a comparison map between thi…
We prove index formulas for elliptic operators acting between sections of C*-vector bundles on a closed manifold. The formulas involve Karoubi's Chern character from K-theory of a C*-algebra to de Rham homology of smooth subalgebras. We show how they apply to the higher index theorem for coverings and to flat foliated bundles, and prove an index theorem for C*-dynamical systems associated to actions of compact Lie groups. In an Appendix we relate the pairing of odd K-theory and KK-theory to the noncommutative spectral flow and prove the regularity of elliptic pseudodifferential operators over …
Eisenstein cocycles in motivic cohomology
本文提出从模曲线同调群到K群同态的新构造，利用GL_2(Z)的1-上闭链直接证明其Hecke等变性。
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