# Extending Intersection Homology Type Invariants to Non-Witt Spaces

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*Markus Banagl*

Intersection homology theory provides a way to obtain
generalized Poincaré duality, as well as a signature and characteristic
classes, for singular spaces. For this to work, one has had to assume however
that the space satisfies the so-called Witt condition. We extend this approach
to constructing invariants to spaces more general than Witt spaces.

We present an algebraic framework for extending generalized
Poincaré} duality and intersection homology to singular spaces
\(X\) not necessarily Witt. The initial step in this program is to
define the category \(SD(X)\) of complexes of sheaves suitable for
studying intersection homology type invariants on non-Witt spaces. The objects
in this category can be shown to be the closest possible self-dual
“approximation” to intersection homology sheaves. It is therefore
desirable to understand the structure of such self-dual sheaves and to isolate
the minimal data necessary to construct them. As the main tool in this analysis
we introduce the notion of a Lagrangian structure (related to the familiar
notion of Lagrangian submodules for \((-1)^k\)-Hermitian forms, as in
surgery theory). We demonstrate that every complex in \(SD(X)\) has
naturally associated Lagrangian structures and conversely, that Lagrangian
structures serve as the natural building blocks for objects in
\(SD(X).\) Our main result asserts that there is in fact an equivalence
of categories between \(SD(X)\) and a twisted product of categories of Lagrangian
structures. This may be viewed as a Postnikov system for \(SD(X)\) whose
fibers are categories of Lagrangian structures.

The question arises as to which varieties possess Lagrangian
structures. To begin to answer that, we define the model-class of varieties
with an ordered resolution and use block bundles to describe the geometry of
such spaces. Our main result concerning these is that they have associated
preferred Lagrangian structures, and hence self-dual generalized intersection
homology sheaves.

#### Table of Contents

# Table of Contents

## Extending Intersection Homology Type Invariants to Non-Witt Spaces

- Contents vii8 free
- Chapter 1. Introduction 112 free
- 1. History 112
- 2. Motivation 112
- 3. The Main Result: A Postnikov System of Lagrangian Structures 213
- 4. Consequences: Characteristic Classes 415
- 5. Ordered Resolutions — A Model Construction 516
- 6. Applications 617
- 7. Further Developments 718
- 8. Sign Questions 718
- 9. Some Remarks on Coefficients 718
- 10. Acknowledgments 819
- 11. Notation 819

- Chapter 2. The Algebraic Framework 920
- Chapter 3. Ordered Resolutions 4960
- Chapter 4. The Cobordism Group Ω[sup(SD)][sub(*)] 5970
- Chapter 5. Lagrangian Structures and Ordered Resolutions 6879
- 1. Statement of Result 6879
- 2. The inductive set–up 6879
- 3. Construction of a nonsingular pairing on H[sup(k)](j*S[sup[.)] 7081
- 4. Stalks of H[sup(k)](j*S[sup[.)] as the hypercohomology of the link of Σ 7182
- 5. The restriction of L[[sup(.)](X[sup((m))]) to V(x) is self–dual 7384
- 6. The construction of a Lagrangian subsheaf of H[sup(k)](j*S[sup[.)] 7485
- 7. The definition of L[sup(.)](X[sup((m+1))]) 7687

- Appendix A. On Signs 7788
- Bibliography 8394