# MA8403 Topics in Algebraic Topology

### Equivariant homotopy theory

This year MA8403 will be on equivariant homotopy theory.

**NB:** For some reason, it may be that you need to sign up with MA8406 instead of MA8403.

### Date and Time

Lectures will be held Wednesdays 12:15 - 14:00 and Fridays 8:15 - 10:00 in MA23. If possible, lectures will be recorded and uploaded to Panopto.

### Exam and grading

Since this is a PhD level course, the course will be pass/fail only. The deliverable (`exam') will be to give a talk of around 30 minutes on a recent(ish) paper in equivariant homotopy theory, or something related to the course. Depending on the number of students/length of the paper, it may be useful to do this in groups of 2 or 3. Here are some suggestions (if you have others, feel free to get in touch). Note that in most cases, there is a specific part of the paper that I have in mind - it's generally not expected that you read and understand the entire paper.

- A new formulation of the equivariant slice filtration with applications to C_p-slices - Hill, Yarnall (would go well with the previous talk)
- An Equivariant Tensor Product on Mackey Functors - Hill, Mazur
- Nilpotence and descent in equivariant stable homotopy theory - Mathew, Naumann, and Noel
- A structure theorem for RO(C2)-graded Bredon cohomology - Clover May (who is located in office 1202 :) )
- The Structure of Mackey Functors - Thevenaz, Webb.

### Lecture Plan

Here is a preliminary plan, which is likely to change.

Semester Week | Topic | Video Link | Comments |
---|---|---|---|

1 | Representation theory and the representation ring | Panopto link for lecture 2 | Worksheet 1 |

2 | Mackey functors | Panopto link for lecture 3 Panopto link for lecture 4 | Worksheet 2 |

3 | G-spaces and equivariant homotopy | Panopto link for lecture 5 Panopto link for lecture 6 | Worksheet 3 |

4 | Elmendorf's theorem and Bredon cohomology | Panopto link for lecture 7 Panopto link for lecture 8 | Worksheet 4 |

5 | No classes | ||

6 | Bredon cohomology continued | Panopto link for lecture 9 Panopto link for lecture 10 | Worksheet 5 |

7 | From spaces to spectra | Panopto link for lecture 11 Panopto link for lecture 12 | Worksheet 6 |

8 | From G-spaces to G-spectra | Panopto link for lecture 13 Panopto link for lecture 14 | Worksheet 7 |

9 | G-spectra part 2 and Q+A Session | Panopto link for lecture 15 Panopto link for lecture 16 | Worksheet 8 |

10 | Fixed point functors | Panopto link for lecture 17 Panopto link for lecture 18 | Worksheet 9 |

11 | Equivariant K-theory, future directions |

### Textbook and References

I will continually be updated a set of lecture notes from the course. The PDF can be found here, while the GitHub repository is at https://github.com/drewheard/MA8403-Notes/