Andrew Ranicki
This paper is an introduction to the use of the cobordism of chain complexes with Poincaré duality in surgery theory. It is a companion to the author's paper "An introduction to algebraic surgery" math.AT/0008071 (to appear in Volume 2 of Surveys in Surgery Theory, Ann. of Maths. Studies, Princeton, 2001) which is an introduction to algebraic surgery using forms and formations.
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The homotopy type of the Poincaré cobordism category for surfaces
本文定义曲面配边范畴的变体，证明其分类空间函子非1-可切除，并给出其一阶导数等价于Thom谱的结果。
Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordis…
We compute the motivic homotopy groups of algebraic cobordism over number fields, the motivic homotopy groups of 2-complete algebraic cobordism over the real numbers and rings of $2$-integers and the motivic homotopy groups of mod 2 motivic Morava $K$-theory over fields with low virtual cohomological dimension. As an application we relate the order of the algebraic cobordism groups of rings of 2-integers to special values of Dedekind $ζ$-functions of totally real abelian number fields.
We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.
Let $X$ be an algebraic variety over $k$ such that $\bar X=X\otimes_k\bar k$ is cellular. We study torsion elements in the Chow ring $CH^*(X)$ which corresponds to $v_iy$ in the algebraic cobordism $Ω^*(\bar X)$ where $0\not=y\in CH^*(\bar X)/p$ and $v_i$ is the generator of $BP^*$ with $|v_i|=-2(p^i-1).$ In particular, we try to compute $CH^*(X)$ from $Ω^*(\bar X)$ when $X$ are twisted complete flag varieties.
We define algebraic cobordism of classifying spaces, Ω^*(BG) and G-equivariant algebraic cobordism Ω^*_G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted F^j(Ω^*(-)), which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We calculate Ω^*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n+1). We also calculate Ω^*(BG) when G is a finite abelian group. A finite non-abelian group for …
In this paper, we study Segre classes in algebraic cobordism. We also prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in the algebraic cobordism of the Grassmannian bundle.
In this paper we study the structure of the Algebraic Cobordism ring of a variety as a module over the Lazard ring, and show that it has relations in positive codimensions. We actually prove the stronger graded version. This extends the result of M.Levine-F.Morel claiming that this module has generators in non-negative codimensions. As an application we compute the Algebraic Cobordism ring of a curve. The main tool is Symmetric Operations in Algebraic Cobordism.
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $\mathbb{C}$ and passing to spaces of $\mathbb{C}$-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we sho…
Poincaré Duality Pairs of $\infty$-Categories
本文引入 $\infty$-范畴对的 Poincaré 对偶概念，统一处理 Wall 吸附、迭代配边等情形，并证明推广的纤维化定理，为后续研究奠定基础。
We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.
This is a sequel to our previous paper of oriented bivariant theory [14]. In 2001 M. Levine and F. Morel constructed algebraic cobordism $Ω_*(X)$ for schemes $X$ over a field $k$ in an abstract way and later M. Levine and R. Pandhairpande reconstructed it more geometrically. In this paper in a similar manner we construct an algebraic cobordism $Ω^*(X \xrightarrow {π_X} S)$ for a scheme $X$ over a fixed scheme $S$ in such a way that if the target scheme $S$ is the point $pt = \operatorname{Spec} k$, then $Ω^{-i}(X \xrightarrow {π_X} pt)$ is isomorphic to Levine-Morel's algebraic cobordism $Ω_i(…
In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Schützenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism.
We construct a cohomology theory using quasi-smooth derived schemes as generators and an analogue of the bordism relation using derived fibre products as relations. This theory has pull-backs along all morphisms between smooth schemes independent of any characteristic assumptions. We prove that in characteristic zero, the resulting theory agrees with algebraic cobordism as defined by Levine and Morel. We thus obtain a new set of generators and relations for algebraic cobordism.
In this article we construct Symmetric operations for all primes (previously known only for p=2). These unstable operations are more subtle than the Landweber-Novikov operations, and encode all p-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map L ---> Z[b_1,b_2,...], providing an important structure on Algebraic Cobordism. Applications include: questions of rationality of Chow group elements - see [11], and the structure of the Graded Algebraic Cobordism. We also construct Steenrod operations of T.tom Dieck-style…
Relying on results of Hopkins-Morel, we show that, for $X$ a quasi-projective variety over a field of characteristic zero, the canonical map $Ω_n(X)\to MGL_{2n,n}'(X)$ is an isomorphism. Here $Ω_*(X)$ is the theory of algebraic cobordism defined by Levine-Morel, and $MGL_{*,*}'$ is the Borel-Moore homology version of the theory of algebraic cobordism defined via the algebraic Thom complex in the Morel-Voevodsky motivic stable homotopy category.
We introduce an equivariant algebraic cobordism theory Ω^G for algebraic varieties with G-action, where G is a split diagonalizable group scheme over a field k. It is done by combining the construction of the algebraic cobordism theory Ωby F. Morel and M. Levine, with the notion of (G, F)-formal group law with respect to a complete G-universe and complete G-flag F as introduced by M. Cole, J. P. C. Greenlees and I. Kriz. In particular, we use their corresponding representing ring L_G(F) in place of the Lazard ring L. We show that localization property and homotopy invariance property hold in Ω…
In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties.
In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration $Ω^*_{(r)}(X)$ are unions of finitely presented $\mathbb{L}$-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules $\mathbb{L}/I(p,n)x$ where $\mathrm{deg\ } x\ge \frac{p^n-1}{p-1}$. As a corollar…
Profinite etale cobordism is a cohomology theory for smooth schemes of finite type over a field. Using an idea of Friedlander, it is constructed as an etale topological analog of the algebraic cobordism theories of Voevodsky and Levine-Morel. With finite coefficients, profinite etale cobordism has a converging Atiyah-Hirzebruch spectral sequence starting from etale cohomology. There are interesting comparison maps from algebraic to profinite etale cobordism. We conjecture that algebraic and etale cobordism with finite coefficients are isomorphic after inverting a Bott element. For the developm…
We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.
We describe additive (unstable) operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence between the set of operations, and the set of transformations: A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying certain simple properties. This provides an effective tool of constructing such operations. As an application, we prove that (unstable) additive operations in Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear c…
Using the work of Dwyer, Weiss, and Williams we associate an invariant to any topologically trivial family of smooth h-cobordisms. This invariant is called the smooth structure class, and is closely related to the higher Franz--Reidemeister torsion of Igusa and Klein. We compute the smooth structure class in terms of a fiberwise generalized Morse function using fiberwise Poincaré--Hopf theory. This computation gives rise to a duality theorem for the smooth structure class that generalizes Milnor's duality theorem for the Whitehead torsion. From this result we deduce a vanishing theorem that im…
An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in arXiv:0709.3905v1 [math.AG]. The algebraic cobordism P^1-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation. For a commutative monoid E in the category SH(k) we identify the set of monoid homomorphisms from MGL to E in the motivic stable homotopy category with the set of all orientations of E. This result was stated originally in a s…
After two papers on weak cubical categories and {\it collarable} cospans, respectively, we put things together and construct a {\it weak} cubical category of cubical {\it collared} cospans of topological spaces. We also build a second structure, called a {\it quasi} cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has {\it lax} identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree.
A bi-variant algebraic cobordism via correspondences
本文通过对应构造了双变元代数配边理论，证明其特例与Levine--Morel及Lee--Pandharipande的代数配边理论同构，是后者的双变元推广。
We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings $M^{2n-1} \subset S^{2n+1}$ under isotopy and cobordism. The second main result (update: which is false) is that the $S$-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi--Levine re…
Slices of the special linear algebraic cobordism spectrum
本文计算了指数特征为e的域F上特殊线性代数配边谱MSL在e可逆化后的切片，结果用特殊酉配边谱的Adams-Novikov谱序列第二页表示，并应用于MSL的切片谱序列以确定其同伦群的前几个Milnor-Witt茎，同时建立了有理特殊线性代数配边谱在任意qcqs概形上的分解。
We describe all operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^* (in the case of a field of characteristic zero). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done earlier, as well as the topological one obtained by T.Kashiwabara. The key new ingredients which permit us to treat the non-additive operations are: the use of "poly-operations" and the "D…
The unit map of the algebraic special linear cobordism spectrum
本文利用框架对应技术计算动机Thom谱的无限$\mathbb{P}^1$-环空间，并证明代数特殊线性配边谱的单位映射在$\mathbb{G}_m$-同伦层上诱导同构。
In this paper, we study the algebraic cobordism spectrum $MSL$ in the motivic stable homotopy category of Voevodsky over an arbitrary perfect field $k$. Using the motivic Adams spectral sequence, we compute the geometric part of the $η$-completion of $MSL$ (modulo the maximal subgroup that is $l$-divisble for all primes $l\neq2, char k$). As an application, we study the Krichever's elliptic genus with integral coefficients, restricted to $MSL$. We determine its image, and identify its kernel as the ideal generated by differences of $SL$-flops. This was proved by B. Totaro in the complex analyt…
We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra $S$ (the Landweber-Novikov algebra) acting on its dual Hopf algebra $S^*$ with a distinguished ``topologically integral'' part $Λ$ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of $S$. We introduce one-dimensional representations of a Hopf algebra. We give series of examples of such representati…
Algebraic cobordism via spans
本文通过跨度范畴定义了配备通用线丛数据的∞-范畴的代数cobordism，恢复了Voevodsky、Gepner和Snaith建立的Thom谱模型，并证明了在满足分裂原理性质的任何定向函子上，投影包公式和相应的Chern类和Whitney和恒等式成立，同时将跨度形式应用于完美oid几何。
On the metalinear algebraic cobordism spectrum
研究金属线性代数配边谱MML的结构，证明其与MSL的等价关系，并计算其Milnor-Witt茎和切片。
The geometric diagonal of the special linear algebraic cobordism
本文构造了c1-球面配边谱的动机版本，通过建立与其他动机Thom谱的联系，计算了局部Dedekind域上特殊线性代数配边同伦群的P1-对角，并讨论了动机Hopf元素η的作用、2-局部化及2-初级挠子群，给出了完全答案。
We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of al…
Nested cobordisms, Cyl-objects and Temperley-Lieb algebras
本文研究嵌套manifolds的离散cobordism类别，通过分层Morse理论生成生成元，并在低维例子Cyl中给出生成元关系。定义Cyl-对象及其与Temperley-Lieb代数等结构的关系，提出新的代数构造。
(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied as a branch of the algebraic and differential topological theory of Morse functions and their higher dimensional versions, or the global singularity theory, by Kalmár, Ikegami, Sadykov, Saeki, Wrazidlo, Yamamoto etc. since 2000s. Cobordism relations are in most cases defined as the following for example; two closed manifolds of a fixed dimension or maps on t…
An interpolation between special linear and general algebraic cobordism $\text{MSL}$ and $\text{MGL}$
本文在稳定动机同伦范畴中建立了 $\text{MSL}$ 与 $\text{MGL}$ 之间的插值关系，由此得到 $\text{MGL}$ 的滤过，并计算了特征零域上 $\text{MSL}$ 的第一切片和稳定同伦群的第一行。
An n-dimensional μ-component boundary link is a codimension 2 embedding of spheres L=\bigsqcup_μS^n \subset S^{n+2} such that there exist μdisjoint oriented embedded (n+1)-manifolds which span the components of L. An F_μ-link is a boundary link together with a cobordism class of such spanning manifolds. The F_μ-link cobordism group C_n(F_μ) is known to be trivial when n is even but not finitely generated when n is odd. Our main result is an algorithm to decide whether two odd-dimensional F_μ-links represent the same cobordism class in C_{2q-1}(F_μ) assuming q>1. We proceed to compute the isomo…
Foams with flat connections and algebraic K-theory
本文提出代数K理论与泡沫配边间的联系，证明环的第一K理论群等同于平面中装饰1-泡沫的配边群，并预期推广到更高维及任意精确范畴。
Chern-Dold character in complex cobordisms and theta divisors
本文用一般主极化阿贝尔簇的光滑 theta 除子表示复配边中 Chern-Dold 特征标的系数，通过引入以对偶 Landweber-Novikov 代数为参数空间的量子化框架解释该特征标，并讨论其与 Milnor-Hirzebruch 问题的联系。
Let f: P-->W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram C<--dT-->P. This homotopy push-out square is an example of what is called a Poincare embedding. We study how to construct algebraic models, in particular in the sense of Sullivan, of that homotopy push-out from a model of the map f. When the codimension is high enough this allows us to completely determine the rational homotopy type of the complement C = W-f(P). Moreover we constru…
We prove that the $\infty$-category of $\mathrm{MGL}$-modules over any scheme is equivalent to the $\infty$-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbb{P}^1$-loop spaces, we deduce that very effective $\mathrm{MGL}$-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential s…
Genera from an algebraic index theorem for supermanifolds
本文证明了 Nest-Tsygan 代数指标定理的超版本，揭示了指标理论、代数变形理论的迹方法与量子场论配分函数中相同的配边不变量，并证明可通过迹方法恢复超对称量子力学中的该不变量。
In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel-Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel-Moore func…
Algebraic cobordism and a Conner-Floyd isomorphism for algebraic K-theory
本文研究了非A^1不变的动机谱稳定∞范畴，利用P^1和加权A^1同伦不变性定义了万有定向动机环谱MGL，并证明了任意qcqs派生概形的代数K理论可通过Conner-Floyd同构从其MGL上同调中恢复。
In the 1980s Matthias Kreck developed a modified surgery theory with obstructions in a hardly understood monoid $l_n(Z[π])$. This paper presents a couple of purely algebraic tools to find out whether an element in $l_{2q}(R)$ is "elementary" i.e. whether a Kreck surgery problem leads to an $h$-cobordism or not.
On the infinite loop spaces of algebraic cobordism and the motivic sphere
本文为域上母题谱MGL、MSL和1的无限环路空间构造了基于Hilbert概形与Quillen加构造的几何模型，并证明正特征下加构造可省略。
We generalise the notion of subdivision of a finite-dimensional locally finite simplicial complex $X$ to geometric algebra, namely to the simplicially controlled categories $\mathbb{A}^*(X)$, $\mathbb{A}_*(X)$ of Ranicki and Weiss. We prove a squeezing result: a bounded chain equivalence of sufficiently algebraically subdivided chain complexes can be squeezed to a simplicially controlled chain equivalence of the unsubdivided chain complexes. Giving $X\times\mathbb{R}$ a bounded triangulation measured in the open cone $O(X_+)$ we use algebraic subdivision to define a functor $\mathrm{"}-\otimes…
Arithmetic field theory via pro-p duality groups
本文通过pro-p对偶群理论定义适合算术拓扑的cobordism范畴，并利用Frobenius代数分类拓扑量子场论，推导出局部p进域上Galois扩展的计数公式。
We factor the virtual Poincare polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as $t^{2u} (t^2-1)^r Q_{G/H}(t^2)$ for a polynomial $Q_{G/H}$ with non-negative integer coefficients. Moreover, we show that $Q_{G/H}(t^2)$ divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.
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