Type
Text
Type
Dissertation
Advisor
Sullivan, Dennis | Morgan, John | Brumfiel, Gregory | Kirillov, Alexander.
Date
2015-12-01
Keywords
Algebraic surgery, E-infinity algebras, E-infinity comodules, Manifolds, Simplicial complexes | Mathematics
Department
Department of Mathematics.
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76403
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using later language, his construction could be said to endow the simplicial chains with an $E_\infty$-coalgebra structure. The second story also begins with a question: when is a space homotopy equivalent to a topological manifold? For dimensions greater than 4, an answer was provided by the work of Browder, Novikov, Sullivan and Wall in surgery theory, which in a later development was algebraically expressed by Ranicki as a single chain level invariant: the total surgery obstruction. After presenting the necessary parts of these stories, the goal of this work will be to express the total surgery obstruction associated to a triangulated space in terms of comodules over the $E_\infty$-coalgebra structure build by Steenrod on its chains. | 74 pages
Recommended Citation
Medina, Anibal, "$E_\infty$-comodules and topological manifolds" (2015). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2326.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2326