Andrew Ranicki
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.
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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…
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…
William Browder in his paper "Surgery and the theory of differentiable transformation groups" developed surgery techniques to study semi-free actions of S1 on homotopy spheres, under the additional assumption that the fixed point set is a homotopy sphere. He used this surgery to show how to construct such actions. In this paper, I discussed a similar theory for semi-free actions of S3 on homotopy spheres. An open problem is raised at the end of the paper.
An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.
An elementary introduction to the principles of algebraic surgery.
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.
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…
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.
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.
The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy class of $h(F)$ is the primary obstruction to desuspending $F$ up to homotopy. The explicit nature of the construction allows for a $π$-equivariant version of $h(F)$ in the case of a $π$-equivariant $F$, with $π$ a discrete group. In earlier joint work we applied the $π_1(N)$-equi…
Interval-sphere model structures
本文提出一种紧致组合模型范畴结构，将区间手术形式化为真正的模型范畴胞腔附着，并揭示余纤维性与持续性理论中驯顺性及代数紧致性的内在联系。
This is a survey paper on spaces of automorphisms of manifolds and spaces of manifolds in a fixed homotopy type. It describes the main theorems of traditional surgery theory, but also the main theorems of pseudoisotopy theory, alias concordance theory, Waldhausen style. It culminates in (an outline of) a synthesis of these two theories, producing algebraic models, valid in a stable range, for spaces of manifolds in a fixed homotopy type. This is inspired by earlier work of Burghelea-Lashof and Hatcher. The algebraic models are a mix of algebraic L-theory and algebraic K-theory.
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 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…
Simple homotopy types of even dimensional manifolds
本文研究偶维闭流形的简单同伦类型，利用代数K理论、手术障碍映射和同伦自同构体构造了首个例子，展示了同伦等价但非简单同伦等价的流形族。
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