---
date: 2023-02-03 00:45
title: Video Global Homotopy Theory
aliases:
  - Video Global Homotopy Theory
annotation-target: 
status: ❌ Not Started
page_current: 0
page_total: 100000
flashcard: 
created: 2023-02-03T00:45
updated: 2024-01-29T17:52
---

Last modified: `=this.file.mday`

# Video Global Homotopy Theory

Reference: https://www.youtube.com/watch?v=EHiZO9-85HY&list=PLsU-oFZDWE_INhBDZSXLQVIa_m0CBb-Ys

Done: 

Lecture 1

- [ ] What is an orthgogonal space?
	- [ ] $X\in \Fun(L, \CGWH)^\cts$ where $L$ is the (topological) linear isometry category (objects = inner product spaces, homs are $\Isom(V, W)$ with the Stiefel manifold topology)
- [ ] What is a continuous functor?
- [ ] What is an orthogonal spectrum?
- [ ] What is the Thom space of a bundle?
	- [ ] Total space of disc bundle mod total space of sphere bundle.
- [ ] What is the orthogonal complement bundle?
	- [ ] $\xi(V, W) \subset W\times L(V\to W)$ given by $\ts{(v, \psi) \st v\perp \psi(V)}$.
- [ ] What is $O(V, W)$?
	- [ ] The one-point compactification of $\xi(V, W)$ the orthogonal complement bundle; equivalently its Thom space.
- [ ] What is the category $O$?
	- [ ] Inner product spaces, homs are $O(V, W)$, composition is the smash product $(w, \phi) \smashprod (v, \psi) \mapsto (w + \phi(v), \phi\circ \psi)$.
- [ ] What is an orthogonal spectrum? A $G\dash$spectrum?
	- [ ] A based $X\in \Fun^\cts(O, \Top_*)$ into based spaces.
	- [ ] For $G\dash$spectra: $X\in \Fun^\cts(O, G\dash\Top_*)$ into based $G\dash$spaces.
- [ ] What is $S^V$ for $V$ a vector space?
	- [ ] $V\union\ts{\infty}$ the one-point compactification.
- [ ] Discuss the actions and structure maps on $\Spectra$.
	- [ ] $O(V, V)  = \Orth(V)_+$ is an orthogonal group and one gets $\Orth(V)_+\smashprod X(V)\to X(V)$
	- [ ] $i_V: S^V \to O(V, V\oplus W)$ induces $\sigma_{V, W}: S^V\smashprod X(W) \to X(V\oplus W)$.
- [ ] How are $G\dash$equivariant homotopy groups defined?
	- [ ] $\pi_0^G(X) \da \colim_{V\in s(U_G)}[S^V\to X(V)]^G_*$ which are $G\dash$equivariant homotopy classes of based maps, and $U_G$ a complete universe of $G\dash$subrepresentations of $U_G$.
	- [ ] $\pi_k^G(X) = \colim_{\cdots}[S^{V\oplus \RR^k}\to X(V)]^G_*$ for $k > 0$ and $\pi_k^G(X) = \colim_{\cdots}[S^{V}\to X(V\oplus \RR^{-k})]^G_*$ for $k < 0$.