Topology of Stratified Spaces: Singular Spaces, Intersection Homology, and Algebraic Geometry by Greg Friedman

Introduction

Singularities appear naturally in many areas of mathematics, from differential geometry to algebraic geometry. For Indian students and researchers seeking a rigorous yet accessible treatment of singular spaces, Topology of Stratified Spaces by Greg Friedman offers a comprehensive and modern perspective. Published by Cambridge University Press, this hardcover volume is an essential resource for advanced undergraduates, graduate students, and professionals working in topology and related fields.

Book Overview

This book provides a deep exploration of stratified spaces—spaces that are not smooth everywhere but can be decomposed into smooth pieces called strata. The author brings together techniques from intersection homology, L2 cohomology, mixed Hodge theory, and characteristic classes to study these singular objects. The text is structured as a collection of expository chapters written by leading experts, making it both a reference work and a gateway to current research. The final chapter presents a list of open problems, offering directions for future investigation.

Key Highlights

• Expert Contributions: Chapters authored by renowned mathematicians specializing in singular spaces and algebraic topology.
• Interdisciplinary Approach: Connects topology, geometry, and analysis through the lens of stratified spaces.
• Open Problems: Includes a curated list of unsolved questions to inspire new research.
• Rigorous Yet Readable: Balances formal proofs with explanatory text suitable for advanced students.

Inside the Book

The volume begins with foundational material on stratified spaces and then progresses to advanced topics such as intersection homology, L2 cohomology, and differential operators on singular varieties. Readers will encounter detailed discussions on signatures and characteristic classes, mixed Hodge theory, and elliptic genera for singular complex and real algebraic varieties. Each chapter is self-contained, allowing readers to focus on specific areas of interest.

Key Topics

• Intersection homology and its applications
• L2 cohomology and differential operators on singular spaces
• Topology of algebraic varieties with singularities
• Signatures and characteristic classes for stratified spaces
• Mixed Hodge theory and its geometric implications
• Elliptic genera of singular complex and real algebraic varieties

Reader Benefits

By engaging with this book, readers will develop a solid understanding of how singularities are studied using modern topological and geometric methods. The clear exposition and expert insights help demystify complex concepts, while the open problems section encourages independent thinking. This is not a beginner's text but rather a valuable companion for those already familiar with basic topology and algebraic geometry.

Learning Outcomes

• Understand the structure and classification of stratified spaces
• Apply intersection homology to compute invariants of singular varieties
• Analyze L2 cohomology and its relationship to differential operators
• Interpret signatures and characteristic classes in the singular context
• Grasp the essentials of mixed Hodge theory and elliptic genera
• Identify open research problems in the topology of singular spaces

Who Should Read

This book is intended for graduate students in mathematics, especially those specializing in topology, algebraic geometry, or differential geometry. It is also an excellent resource for researchers and faculty members who wish to update their knowledge of stratified spaces. Advanced undergraduates with a strong background in algebraic topology will find it challenging but rewarding.

About the Author

Greg Friedman is a professor of mathematics known for his contributions to the topology of singular spaces and intersection homology. His research bridges algebraic topology and geometry, and he has authored numerous papers on stratified spaces, characteristic classes, and related topics. His expository style in this volume reflects his commitment to making advanced mathematics accessible.

About the Publisher

Cambridge University Press is one of the world's oldest and most respected academic publishers. With a strong tradition of publishing high-quality mathematics texts, Cambridge ensures that this hardcover edition meets the rigorous standards expected by scholars and students alike. The press is known for its authoritative titles in science and mathematics.

Conclusion

Topology of Stratified Spaces is a definitive work that brings together cutting-edge research and foundational theory. For Indian readers pursuing advanced studies in topology or algebraic geometry, this book is a valuable investment. Its hardcover binding ensures durability, and the content remains relevant for years to come. Add it to your library to deepen your understanding of singular spaces and their role in modern mathematics.