Topics in Homotopy Theory and Topology
This is the second semester of the course Topics in Modern Homotopy Theory and I will suppose familiarity with the basic contents of that course. This includes some knowledge of point-set topology and basic algebraic topology. Lecture notes from last semester are available, and it is certainly not necessary to have completed that course to attend this one.
We begin the course by studying fibrations in Top. Hurewicz and Serre fibrations are introduced and their basic properties compared. Time permitting we will also discuss quasifibrations and Dold fibrations. From here, homotopy pullback squares are introduced. The first major result of the course will be to state and prove the Mather Cube Theorems. These have many applications, and time permitting we will use them to study the Lusternik-Schnirelmann category and topological complexity, as well as to prove the Bott-Samelson Theorem.
For the next part of the course we focus on categorical methods. We introduce model categories and use the material from the first part of the course to establish the three main model structures on the category of topological spaces (these being the Strøm, Quillen, and mixed model structures). Elements of categorical topology will be presented, and this will lead us to introduce several convenient categories of topological spaces, including compactly generated, sequential, and numerically generated spaces.
In the third section of the course we will formally introduce CW complexes. Although the audience is expected to be familiar with the basic definitions, our goal is to establish some of their more advanced properties. An important property is their stratifiability, which implies their paracompactness and perfect normality. Applications will be given to their metrisability and products. Simplicial complexes will also be defined and studied. These constitute a subclass of CW complexes whose convenient properties will be used in the last section of the course.
Finally, we give the metric characterisation of CW topology. For this we need to introduce absolute neighbourhood retracts (ANRs) and prove a classic theorem due to Milnor and Hanner, that a space has CW homotopy type if and only if it has ANR homotopy type. Time permitting we will prove a more recent and less well kown result due to R. Cauty, which states that a metric space is an ANR if and only if it each of its open subsets is homotopy equivalent to a CW complex. We end the course by giving applications to the Dranishnikov-Dydak homotopical approach to dimension and cohomological dimension of metric spaces.
We begin the course by studying fibrations in Top. Hurewicz and Serre fibrations are introduced and their basic properties compared. Time permitting we will also discuss quasifibrations and Dold fibrations. From here, homotopy pullback squares are introduced. The first major result of the course will be to state and prove the Mather Cube Theorems. These have many applications, and time permitting we will use them to study the Lusternik-Schnirelmann category and topological complexity, as well as to prove the Bott-Samelson Theorem.
For the next part of the course we focus on categorical methods. We introduce model categories and use the material from the first part of the course to establish the three main model structures on the category of topological spaces (these being the Strøm, Quillen, and mixed model structures). Elements of categorical topology will be presented, and this will lead us to introduce several convenient categories of topological spaces, including compactly generated, sequential, and numerically generated spaces.
In the third section of the course we will formally introduce CW complexes. Although the audience is expected to be familiar with the basic definitions, our goal is to establish some of their more advanced properties. An important property is their stratifiability, which implies their paracompactness and perfect normality. Applications will be given to their metrisability and products. Simplicial complexes will also be defined and studied. These constitute a subclass of CW complexes whose convenient properties will be used in the last section of the course.
Finally, we give the metric characterisation of CW topology. For this we need to introduce absolute neighbourhood retracts (ANRs) and prove a classic theorem due to Milnor and Hanner, that a space has CW homotopy type if and only if it has ANR homotopy type. Time permitting we will prove a more recent and less well kown result due to R. Cauty, which states that a metric space is an ANR if and only if it each of its open subsets is homotopy equivalent to a CW complex. We end the course by giving applications to the Dranishnikov-Dydak homotopical approach to dimension and cohomological dimension of metric spaces.
Lecturer
Tyrone Cutler
Date
20th September ~ 14th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Thursday | 15:20 - 17:50 | A3-3-301 | ZOOM 09 | 230 432 7880 | BIMSA |
Syllabus
(1) Analysis of Cofibrations and Fibrations in Top
The homotopy lifting property. Hurewicz fibrations, Serre fibrations.
Mutual characterisation of fibrations and cofibrations by orthogonality.
The homotopy category is a homotopy category.
Locally trivial maps. Introduction to fibrewise methods.
(2) Basic homotopy limits.
Homotopy pullbacks.
The Mather Cube Theorem.
Applications to the Bott-Samelson Theorem.
(3) Further Topics.
Elements of Lusternik-Schnirelmann category and topological complexity.
H-spaces and co-H-spaces.
Categorical methods (homotopical categories and model categories).
Convenient categories of topological spaces.
(4) CW Topology
Adjunction spaces and towers of spaces.
Topological properties and metrisation of CW complexes. Products of CW complexes.
Homotopical aspects of CW topology (the Whitehead Theorem).
Simplicial complexes. Polyhedra.
(5) Partitions of unity in homotopy theory
Numerable covers. Paracompactness.
The local nature of fibrations and cofibrations.
The nerve of an open cover.
ˇCech extension of homotopy functors.
(6) Extension Theory
Absolute neighbourhood retracts and extensors.
The metric characterisation of CW topology
Applications to function space topology.
The Dranishnikov-Dydak approach to dimension and cohomological dimension.
The homotopy lifting property. Hurewicz fibrations, Serre fibrations.
Mutual characterisation of fibrations and cofibrations by orthogonality.
The homotopy category is a homotopy category.
Locally trivial maps. Introduction to fibrewise methods.
(2) Basic homotopy limits.
Homotopy pullbacks.
The Mather Cube Theorem.
Applications to the Bott-Samelson Theorem.
(3) Further Topics.
Elements of Lusternik-Schnirelmann category and topological complexity.
H-spaces and co-H-spaces.
Categorical methods (homotopical categories and model categories).
Convenient categories of topological spaces.
(4) CW Topology
Adjunction spaces and towers of spaces.
Topological properties and metrisation of CW complexes. Products of CW complexes.
Homotopical aspects of CW topology (the Whitehead Theorem).
Simplicial complexes. Polyhedra.
(5) Partitions of unity in homotopy theory
Numerable covers. Paracompactness.
The local nature of fibrations and cofibrations.
The nerve of an open cover.
ˇCech extension of homotopy functors.
(6) Extension Theory
Absolute neighbourhood retracts and extensors.
The metric characterisation of CW topology
Applications to function space topology.
The Dranishnikov-Dydak approach to dimension and cohomological dimension.
Reference
M. Arkowitz, Introduction to Homotopy Theory.
J. Strom, Modern Classical Homotopy Theory.
A. Hatcher, Alegbraic Topology.
K. Sakai, Geometric Aspects of General Topology.
E. Riehl, Categorical Homotopy Theory.
J. Strom, Modern Classical Homotopy Theory.
A. Hatcher, Alegbraic Topology.
K. Sakai, Geometric Aspects of General Topology.
E. Riehl, Categorical Homotopy Theory.
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English