Yu. V. Muranov

SPLITTING ALONG SUBMANIFOLDS AND L-SPECTRA

AbstractThe problem of splitting of homotopy equivalence along a submanifold is closely related to surgeries of submanifolds and exact sequences in surgery theory. We describe possibilities and methods of application of

Splitting Obstruction Groups in Codimension 2

\mathbb{L}

On iterated Browder-Livesay invariants

-spectra for the investigation of the problem of splitting of (simple) homotopy equivalence of manifolds along submanifolds. This approach naturally leads us to commutative diagrams of exact sequences, which play an important role in calculational problems of surgery theory.

The π-π-theorem for manifold pairs with boundaries

In 1978 Cappell and Shaneson pointed out interesting properties of the Browder–Livesay invariants, which are analogous to the differentials of a certain spectral sequence. Such a spectral sequence was constructed by Hambleton and Kharshiladze in 1991. The main step of the construction of the spectral sequence consists in constructing an infinite filtration of spectra, in which, as is well known, only the first two spectra have a clear geometric meaning. In the present paper a geometric interpretation is given to all the spectra of the filtration in the Hambleton–Kharshiladze construction. Surgery obstruction groups for a system of embedded manifolds are introduced, and it is proved that the spectra realizing these groups coincide with the spectra in the Hambleton–Kharshiladze filtration. The algebraic and geometric properties of these groups and their connections with classical surgery theory are described. An isomorphism between these groups and the Browder–Quinn surgery obstruction groups for stratified manifolds is established. The results obtained are applied to the problem of realization of elements of the Wall groups by normal maps of closed manifolds and to the study of the iterated Browder–Livesay invariants.

Surgery on triples of manifolds

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.

Normal invariants of manifold pairs and assembly maps

In the paper the obstruction groups to obtaining simple homotopy equivalence by surgery from normal degree 1 maps of closed manifolds with dihedral fundamental group are computed. The cases of trivial orientation for the dihedral group and nontrivial orientation for the order 2 cyclic subgroup are considered. New results concerning the Browder-Livesey groups and natural maps ofL-groups arising in index 2 inclusions of the cyclic group into the dihedral group are obtained.

Transfer maps for triples of manifolds

The Browder-Livesay invariants provide obstructions to the realization of elements of Wall groups by normal maps of closed manifolds. A generalization of the iterated Browder-Livesay invariants is proposed and properties of the invariants obtained are described. The generalized definition makes it possible to investigate the relationship between a normal map and its restriction to a submanifold and clarifies the relationship between the Browder-Livesay invariants and the Browder-Quinn groups of obstructions to surgery on filtered manifolds. Several theorems describing a relationship between a normal map and its restriction to a submanifold are proved.

RELATIVE WALL GROUPS AND DECORATIONS

The surgery obstruction of a normal map to a simple Poincaré pair (X, Y) lies in the relative surgery obstruction group L*(π1(Y) → π1(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 Poincaré pair with π1(X) ≅ π1(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.