Topological manifolds

This is a course on the theory of topological manifolds.
Here is the syllabus.
The course is aimed at master's students, graduate students, and researchers who are not experts.
The prerequisites are point set topology and the basics of algebraic topology.


A manifold is a Hausdorff, paracompact topological space that is locally homeomorphic to Euclidean space. One frequently encounters manifolds with additional structure, such as smooth, piecewise-linear (PL), Riemannian, or symplectic. The aim of this course is to learn about unadulterated topological manifolds. For such manifolds, many essential tools from the start of differential topology are deep theorems. Examples include the existence and uniqueness of collar neighbourhoods for boundaries of manifolds, the existence of tubular neighbourhoods, handlebody structures, and transversality for submanifolds. We will learn about these results, and what goes into their proofs, for topological manifolds.


Lectures will be streamed through zoom. Recordings will be made available afterwards. Bonn participants: check eCampus, others please email us for access to the recordings.
Zoom link. Password: torustrick
Wednesdays and Thursdays 1015-1200, German time, 28th October 2020 - 11th February 2021.
Contact details:

Lecture notes

Version from 15th March 2021.
This is the final version of the course notes.
Thanks to Gopal Ananthakrishna, Anthony Conway, Ekin Ergin, Danica Kosanovic, Christian Kremer, Isacco Nonino, and Ben Ruppik for typing, editing, drawing pictures, and providing solutions to exercises.

Problem sheets

Problem Sheet 1.
Problem Sheet 2.
Problem Sheet 3.
Problem Sheet 4.
Problem Sheet 5.
Problem Sheet 6.
Problem Sheet 7.
Problem Sheet 8.
Problem Sheet 9.
Problem Sheet 10.
Problem Sheet 11.

Summer semester seminar

We will run a seminar course in the summer semester 2021, in which participants give talks. Notes from the talks will contribute to the lecture notes project.


Here is a collection of references and links.

General resources
Rob Kirby's lectures, Edinburgh, November 2012.
Andrew Ranicki's collection of references.
Foundational essays on topological manifolds, smoothings, and triangulations by R. Kirby and L. Siebenmann.
Geometric topology notes by S. Ferry.
Recent advances in topological manifolds by A. Casson.
Original handwritten version of Casson's lecture notes by A. Ranicki.
Fragments of Geometric Topology from the Sixties by S. Buonchristiano.
Lecture notes on topological manifolds by A. Kupers. An extended version can be found in here.
Counting topological manifolds by J. Cheeger and J. Kister.

Wild embeddings
Some wild cells and spheres in three-dimensional space by E. Artin and R. Fox.
Embeddings in manifolds by R. Daverman and G. Venema.
Topological embeddings by T. B. Rushing.

Collars on boundaries
Locally Flat Imbeddings of Topological Manifolds by M. Brown.
A new proof of Brown's collaring theorem by R. Connelly.
Collars and concordances of topological manifolds by M. Armstrong.

Schoenflies theorem
A proof of the generalized Schoenflies theorem by M. Brown.
The canonical Schoenflies theorem by D.B. Gauld.

Microbundles by J. Milnor.
Microbundles and differentiable structures by J. Milnor.
Microbundles are fibre bundles by J. Kister.
On normal microbundles by M. Hirsch.
An embedding without a normal microbundle by C.P. Rourke and B.J. Sanderson.

Torus trick
Stable homeomorphisms and the annulus conjecture by R. Kirby.
Deformations of Spaces of Imbeddings by R. Edwards and R. Kirby.
The Kirby Torus Trick for Surfaces by A. Hatcher.
The triangulation of 3-manifolds by A.J.S. Hamilton.
On bundles over a sphere with fibre Euclidean space by C.T.C. Wall.
Structures on MxR by W. Browder.
Handles, semihandles, and destabilization of isotopies by D. Webster.

Explicit codimension zero immersions of the punctured torus in Euclidean space: by Ferry, Milnor, and Barden.
La classification des immersions combinatoires by A. Haefliger and V. Poénaru.
Immersions and surgeries on topological manifolds by J.A. Lees.
Lees' immersions theorem and the triangulation of manifolds by R. Lashof.
Algebraic Topology proceedings 1971 by A. Liulevicious (ed). See p.131 The immersion approach to triangulation and smoothing by R. Lashof.
Algebraic Topology proceedings Aarhus 1970. See p. 282, The immersion approach to triangulation and smoothing by R. Lashof.
Groups of Automorphisms of Manifolds by D. Burghelea, R. Lashof, and M. Rothenberg. Appendix by E. Pedersen on the topological category.

Homeomorphism groups
Local contractibility of the group of homeomorphisms of a manifold by A.V. Cernavskii.
Homotopy in homeomorphism spaces, TOP and PL by M.-E. Hamstrom.

Homotopy tori
On homotopy tori and the annulus theorem by C.T.C. Wall.
On homotopy tori II by W.C. Hsiang and C.T.C. Wall.
Fake tori, the annulus conjecture, and the conjectures of Kirby by W.C. Hsiang and J. Shaneson.
Fake tori by W.C. Hsiang and J. Shaneson.
The topological rigidity of the torus, MSc thesis of Alexandre Martin.

Stable structures
Stable structures on manifolds by M. Brown and H. Gluck.
Stable structures on manifolds: I Homeomorphisms of Sn by M. Brown and H. Gluck.
Stable structures on manifolds: II Stable manifolds by M. Brown and H. Gluck.
Stable structures on manifolds: III Applications by M. Brown and H. Gluck.

Piecewise Linear manifolds
Seminar on combinatorial topology by C. Zeeman.
Piecewise linear topology by J.F.P. Hudson.
Introduction to piecewise-linear topology by C. Rourke and B. Sanderson.
Lectures on polyhedral topology by J. Stallings.
Variétés linéaires par morceaux et variétés combinatoires by C. Dedecker.
Microbundles and bundles I by N. Kuiper and R. Lashof.
Microbundles and bundles II by N. Kuiper and R. Lashof.
Microbundles and smoothing by R. Lashof and M. Rothenberg.
Block bundles: I by C. Rourke and B. Sanderson.
Geometric topology notes by J. Lurie.

Smoothing and PLing theory
Notes on Smoothing theory by J. Davis.
Manifolds and Poincaré complexes by J. Davis and N. Petrosyan.
Piecewise linear structures on topological manifolds by Y. Rudyak.
Triangulations of manifolds I by R. Lashof and M. Rothenberg.
The Hauptvermutung book by A. Ranicki (ed), A. Casson, D. Sullivan, M. Armstrong, C. Rourke, and G. Cooke.
Manifolds - Amsterdam 1970 by N. Kuiper (ed). See first 5 papers.
Topology of manifolds - Proceedings of Georgia Topology conference 1969 by J. Cantrell and C. Edwards (ed). See Chapter 1.

Structures on Euclidean space and the Poincaré conjecture
The piecewise linear structure of Euclidean space by J. Stallings.
Finding a boundary for an open manifold by W. Browder, J. Levine, and G. Livesay.
Polyhedral homotopy spheres by J. Stallings.
The generalised Poincaré conjecture by E. C. Zeeman.
Generalised Poincare's conjecture in dimensions greater than four by S. Smale.
Differentiable and combinatorial structures on manifolds by S. Smale.
The engulfing theorem for topological manifolds by M. H. A. Newman.