Splitting Obstruction Groups Research Articles

Splitting Obstruction Groups Along one-Sided Submanifolds

We construct new commutative diagrams of exact sequences which relate surgery and splitting obstruction groups for pairs of manifolds. The splitting and surgery obstruction groups are computed for pairs of manifolds and various geometric diagrams of groups corresponding to the problem of splitting along a one-sided submanifold of codimension 1.

Browder-Livesay Filtrations and the Example of Cappell and Shaneson

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 : M 3 × T 2 → M 3 × 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 \({\sigma (f) \in L_{5}(\pi _{1}(M))}\) is based on consideration of a Browder-Livesay filtration of a manifold X with \({\pi _{1}(X) \cong \pi _{1}(M)}\) . For a Browder-Livesay pair \({Y^{n-1} \subset X^{n}}\), the restriction of a normal map to the submanifold Y is given by a partial multivalued map Γ : L n (π 1(X)) → L n−1(π 1(Y)), and the Browder-Livesay filtration provides an iteration Γn. This map is a basic step in the definition of the iterated Browder-Livesay invariants which give obstructions to realization of surgery obstructions by normal maps of closed manifolds. In the present paper we prove that Γ 3(σ (f)) = 0 for any Browder-Livesay filtration of a manifold X 4k+1 with \({\pi _{1}(X) \cong Q}\) . We compute splitting obstruction groups for various inclusions ρ → Q of index 2, describe natural maps in the braids of exact sequences, and make more precise several results about surgery obstruction groups of the group Q.

The π-π-theorem for manifold pairs with boundaries

The surgery obstruction of a normal map to a simple Poincare pair (X, Y) lies in the relative surgery obstruction group L *(π 1(Y) → π 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) ≅ π 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 the surgery obstruction groups LP * for manifold pairs and splitting obstruction groups LS *. In the present paper, we formulate and prove for manifold pairs with boundary results similar to the π-π-theorem. We give direct geometric proofs, which are based on the original statements of Wall’s results and apply obtained results to investigate surgery on filtered manifolds.

On the splitting problem for manifold pairs with boundaries

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 $(Y, \partial Y)\subset (X, \partial X)$ with boundary which is a simple homotopy equivalence on the boundary $\partial X$. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups of obstructions to splitting of such mixed structures along submanifold with boundary $(Y, \partial Y)$. We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric examples.

Relative groups in surgery theory

In this paper we consider various types of relative groups which naturally arise in surgery theory, and describe algebraic properties of them. Then we apply the obtained results to investigate the splitting obstruction groups $LS_*$ and the surgery obstruction groups $LP_*$ for a manifold pair. Finally, we introduce the lower $LS_*$- and $LP_*$-groups, and describe connections between them and the corresponding lower $L_*$-groups and surgery exact sequence.

Splitting Obstruction Groups in Codimension 2

The splitting obstruction groups depend functorially on the square of fundamental groups. In the paper the problem of splitting along a submanifold of codimension two under some restrictions on the square of fundamental groups is considered. New exact sequences and commutative diagrams containing Wall groups, splitting obstruction groups, and surgery obstruction groups for manifold pairs are obtained. Examples of computation of splitting obstruction groups and natural maps are considered.

The Groups LS and Morphisms of Quadratic Extensions

For a morphism of quadratic extensions of antistructures, groups similar to the groups of obstructions to splitting along one-sided submanifolds are defined. These groups are a natural generalization of the splitting obstruction groups. The results obtained open additional possibilities for constructing groups and natural maps in L-theory.

Projective splitting obstruction groups for one-sided submanifolds

A geometric diagram of groups, which consists of groups equipped with geometric antistructures, is a natural generalization of the square of fundamental groups arising in the splitting problem for a one-sided submanifold. In the present paper the groups and of such diagrams are defined and the properties of these groups are described. Methods for the computation of , -groups and natural maps in diagrams of exact sequences are developed in the case of a geometric diagram of finite 2-groups.

Groups of obstructions to surgery and splitting for a manifold pair

The surgery obstruction groups of manifold pairs are studied. An algebraic version of these groups for squares of antistructures of a special form equipped with decorations is considered. The squares of antistructures in question are natural generalizations of squares of fundamental groups that occur in the splitting problem for a one-sided submanifold of codimension 1 in the case when the fundamental group of the submanifold is mapped epimorphically onto the fundamental group of the manifold. New connections between the groups , the Novikov-Wall groups, and the splitting obstruction groups are established.

Splitting obstruction groups and quadratic extensions of anti-structures

A natural generalization is given of the LS groups of obstructions to splitting in the case of one-sided manifolds. These LS groups depend functorially on four anti-structures and maps between them, generating a square of a special form. The connection between the LS groups and the Wall groups of anti-structures involved in the diagram is studied. “Intermediate” LS groups (LS with decorations) are also defined, by analogy with the corresponding Wall groups, and the connection between LS groups with different decorations is studied.