Andrew Ranicki
An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.
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The Wall surgery obstruction groups have two interesting geometrically defined subgroups, consisting of the surgery obstructions between closed manifolds, and the inertial elements. We show that the inertia group $I_{n+1}(π,w)$ and the closed manifold subgroup $C_{n+1}(π,w)$ are equal in dimensions $n+1\geq 6$, for any finitely-presented group $π$ and any orientation character $w\colon π\to \cy 2$.
Closed manifold surgery obstructions and the Oozing Conjecture
本文通过新例子反驳了渗出猜想，揭示了闭流形在高维中的非平凡手术障碍特性。
The total surgery obstruction of a finite n-dimensional Poincare complex X is an element s(X) of a certain abelian group S_n (X) with the property that for n >= 5 we have s(X) = 0 if and only if X is homotopy equivalent to a closed n-dimensional topological manifold. The definitions of S_n (X) and s(X) and the property are due to Ranicki in a combination of results of two books and several papers. In this paper we present these definitions and a detailed proof of the main result so that they are in one place and we also add some of the details not explicitly written down in the original source…
This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $σ^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds of dimension $n \geq 5$. Our proof uses essentially the geometrically defined $\mathbb{L}$-spectrum as described by Nicas (going back to Quinn) and some well known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describ…
Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $LP_*$ and splitting obstruction groups $LS_*$. In the present paper we formulate and …
Cappell and Shaneson pointed out in 1978 interesting properties of Browder - Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of Wall groups by normal maps of closed manifolds. The main step of construction of the spectral sequence is an infinite filtration of spectra in which only the first two, as is well-known, have clear geometric sense. The first one is a spectrum $\Bbb L(π_1(X))$ for surgery obstruc…
The paper introduces a group $LSP$ of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the $LSP$-groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery obstruction group of the ambient manifold to the $LSP$-group provides an invariant when elements of the Wall group are not realized by normal maps of closed manifolds. Some $LSP$-groups are computed precisely.
An elementary introduction to the principles of algebraic surgery.
The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $ rel \partial$ case the surgery on a manifold pair is considered with the given fixed manifold structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps on the boundary. Consider a normal map to a manifold pair $(Y, \partial Y)\subset (X, \partial …
Browder-Novikov-Sullivan-Wall surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory (such as the Wall surgery obstruction groups), without losing sight of the geometric motivation.
It is well-known that an n-dimensional Poincaré complex $X^n$, $n \ge 5$, has the homotopy type of a compact topological $n$-manifold if the total surgery obstruction $s(X^n)$ vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic and controlled surgery theory. Next, we discuss the key idea of Quinn's approach. Finally, we present some cases of special fundamental groups, due to the authors and to Yamasaki.
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.
Let $M^3$ be a 3-dimensional manifold with fundamental group $π_1(M)$ which contains a quaternion subgroup $Q$ of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map $ f\colon M^3\times T^2\to M^3\times S^2$ which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of $σ(f)\in L_5(π_1(M))$ is based on consideration of a Browder-Livesay filtration of a manifold $X$ with $π_1(X)\cong π_1(M)$. For a Browder-Livesay pair $Y^{n-1…
The inertia subgroup $I_n(π)$ of a surgery obstruction group $L_n(π)$ is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $π_1(X)=π$. This group is a subgroup of the group $C_n(π)$ which consists of the elements which can be realized by normal maps of closed manifolds. In all known cases these groups coincide and the computation of them is one of the basic problems of surgery theory. The computation of the group $C_n(π)$ is equivalent to the computation the image of the assembly map $A:H_{n}(Bπ, \bol…
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.
Wall's finiteness obstruction is an algebraic K-theory invariant which decides if a finitely dominated space is homotopy equivalent to a finite CW complex. The object of this survey is to describe the invariant (which was first formulated in 1965) and some of its many applications to the surgery classification of manifolds.
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…
Homology manifolds and euclidean bundles
构造一个庞加莱复形，其周期性全手术障碍消失但斯皮瓦克法丛不能约化为稳定欧几里得丛，以此反驳文献中关于相关庞加莱复形同伦等价于同调流形及同调流形斯皮瓦克法丛可约化的两个断言。
The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into $R^{2n}$ for $n\neq 2$, and it was recently shown to be incomplete for $n=2$. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of 2-complexes in $R^4$.
Algebraic criteria for stable diffeomorphism of spin 4-manifolds
本文研究闭连通自旋4-流形的稳定微分同胚分类，提出次障碍和三级障碍与代数不变量的关联猜想，对多类基本群证明了猜想，并给出基本群为$\mathbb{Z} \times \mathbb{Z}/2$的自旋4-流形的完整代数稳定分类。
We give a numerical algorithm computing Euler obstruction functions using maximum likelihood degrees. The maximum likelihood degree is a well-studied property of a variety in algebraic statistics and computational algebraic geometry. In this article we use this degree to give a new way to compute Euler obstruction functions. We define the maximum likelihood obstruction function and show how it coincides with the Euler obstruction function. With this insight, we are able to bring new tools of computational algebraic geometry to study Euler obstruction functions.
Cappell's codimension 1 splitting obstruction surgery group UNil_n(R;R,R) of a ring with involution R is a direct summand of the Wall surgery obstruction group L_n(R[D_{\infty}]) of the amalgamated free product R[D_{\infty}] = R[Z_2]*_RR[Z_2], with D_{\infty}=Z_2*Z_2 the infinite dihedral group. We use the quadratic Poincaré cobordism formulation of the L-groups to prove that L_n(R[x]) = L_n(R)\oplus UNil_n(R;R,R), with \bar{x} = x . We combine this with M. Weiss' universal chain bundle theory to produce almost complete calculations of UNil_*(Z;Z,Z) and L_*(Z[D_{\infty}]).
We compare two properties for a CW-space $X$ of finite type: (1) being homotopy equivalent to a CW-complex without $j$-cells for $k\leq j\leq \ell$ (($k,\ell$)-cellfree) and (2) $H^j(X;R)=0$ for any $\mathbb Zπ_1(X)$-module $R$ when $k\leq j\leq \ell$ (cohomogy ($k,\ell$)-silent). Using the technique of Wall's finiteness obstruction, we show that a connected CW-space $X$ of finite type which is cohomogy ($k,\ell$)-silent determines a "cell-dispensability obstruction'' $w_k(X)\in\tilde K_0(\mathbb Zπ_1(X))$ which vanishes if and only if $X$ is ($k,\ell$)-cellfree ($k\geq 4$). Any class in $\til…
Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more general result about iterated group extensions. As an application, we obtain new criteria for formality of spaces, and 1-formality of groups, illustrated by bundle constructions and various examples from low-dimensional topology and singularity theory.
This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $RP^2$, which reports the location of the essential 1-cycle. In contrast, we show that there exists an uncountable collection of topological arbiters in dimension 4. Families of arbiters, not induced by homology, are also shown to exist in higher dimensions. The technical ingredie…
Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, i.e., lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension $\leq 3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, w…
We derive rational (Sullivan) models for configuration spaces of points on manifolds purely from algebraic considerations via obstruction theory, essentially without the use of analytic or geometric techniques.
We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.
This is a survey of the current state of the question "Which closed connected manifolds of dimension $n\ge 5$ admit Riemannian metrics whose scalar curvature function is everywhere positive?" The introduction gives a brief overview of these results, while the body of the paper discusses the methods used in the proofs of these results. We mention the two flavors of topological obstructions to the existence of \pscm s: one is a consequence of the Weizenböck formula for the Dirac operator, the other is obtained by considering stable minimal hypersurfaces. We talk about geometric constructions of …
Obstruction sequences to homotopy equivalences
研究完备微分分次李代数中规范等价存在性的阻碍理论，用于刻画特定微分分次代数间同伦等价，应用此理论在代数拓扑和几何中建立新结果，尤其关注高连通簇极小模型。
We define a variant of intersection space theory that applies to many compact complex and real analytic spaces $X$, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions", which we can choose to vanish, and verify that algebra…
In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an example of triangulated category without models. We also obtain cohomological characterizations of pre-triangulated DG, A-infinity, and spectral categories.
A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lov% ász \cite{Lovasz} through the solution of the Lovász conjecture \cite% {Babson-Kozlov}, \cite{Carsten}, many methods from Algebraic Topology have been used. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. For example, the following problem…
The structure space S(M) of a closed topological m-manifold M classifies bundles whose fibers are closed m-manifolds equipped with a homotopy equivalence to M. We construct a highly connected map from S(M) to a concoction of algebraic L-theory and algebraic K-theory spaces associated with M. The construction refines the well-known surgery theoretic analysis of the block structure space of M in terms of L-theory.
$K$-Theoretic Obstructions to Linearizing QCA Representations
本文针对量子元胞自动机表示，利用代数$K$-理论谱发展障碍理论，研究其线性化问题，并计算了点、线和平面上QCA空间的同伦类型。
Kaledin classes and formality criteria
本文发展了任意交换基环上代数结构形式性的一般障碍理论，构造了忠实检测微分分次代数形式性的Kaledin障碍类，推广了先前理论并建立了多种新颖的形式性准则。
For any two graphs $G$ and $H$ Lovász has defined a cell complex $Hom(G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with $G$ a cycle of odd length. More specifically, we show that: if $Hom(C_{2r+1},G)$ is $k$-connected, then $χ(G)\geq k+4$. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney classes.
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