---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: 4-manifolds
aliases: [4-manifold, "four manifolds", "4-manifold", "4-manifolds"]

---

Last modified date: <%+ tp.file.last_modified_date() %>


---

- Tags 
	- #geomtop 
- Refs:
	- [Classification of 4-manifolds](https://www.youtube.com/watch?v=yhbjlMyX0Yk) #resources/videos 
	- Stanford course notes: <https://stanford.edu/~sfh/283A.pdf#page=2> #resources/course-notes 
- Links:
	- [Unsorted/Rohklin's theorem](Unsorted/Rohklin's%20theorem.md)
	- [Unsorted/Kodaira-Spencer](Unsorted/Kodaira-Spencer.md)
	- [Unsorted/riemannian manifold](Unsorted/riemannian%20manifold.md)
	- [Unsorted/intersection form](Unsorted/intersection%20form.md)
	- [Unsorted/Rokhlin theorem](Unsorted/Rokhlin%20theorem.md)

---

# 4-Manifolds

Relation to [[11 8 Conjecture]], [Poincaré](Unsorted/Poincare%20conjectures.md), and [Ricci flow](Unsorted/Ricci%20flow.md). See also [minimal model program](Unsorted/minimal%20model%20program.md) and [[abundance conjecture]].

![](attachments/Pasted%20image%2020221231200100.png)

There exist (simply connected) closed oriented 4-manifolds which are homeomorphic but not diffeomorphic
![](attachments/Pasted%20image%2020220422095025.png)

# Review Terms

#projects/review #projects/notes/reading #todo/questions 

-  What is the [tautological bundle](tautological%20bundle.md)?
-  What is [Serre duality](Serre%20duality.md)?
-  What is [Hirzebruch-Riemann-Roch](Hirzebruch-Riemann-Roch.md)?
- How is $\mathcal{O}(n)$ defined?
- What is a [Kahler manifold](Kahler%20manifold)?
- What is $A^{p, q}$?
- How is [Dolbeaut cohomology](Dolbeaut%20cohomology.md) defined?
- What is [Noether's formula](Noether's%20formula)?
- What is the [symbol](symbol.md) of an operator?
- What is an [elliptic complex](elliptic%20complex.md)?
- What is the [Aatiyah-Singer Index Theorem](Aatiyah-Singer%20Index%20Theorem.md)?
- What is the [Chern class](Chern%20class.md)?
- What is the [Chern character](Chern%20character)?
- What are [Chern roots](Chern%20roots)?
- What is the [Todd class](Todd%20class)?
- What is the [Euler class](Euler%20class.md)?
- What is the [Laplacian](Laplacian.md)?
- What is a [Hermitian metric](Hermitian%20metric)?
- What is the [Hodge star](Hodge%20star.md)?
- What is the [genus formula](genus%20formula)?
- What is the [blowup](blowup.md)
	- What is the [exceptional divisor](exceptional%20divisor)?
	- What is the [strict transform](strict%20transform)?
	- What is the [blowdown](blowdown)
- What is a [unimodular lattice](unimodular%20lattice)?
- What is the [Steifel-Whitney class](Steifel-Whitney%20class)?
- What is the [Pontrayagin class](Pontrayagin%20class.md)?
- What is a [Kahler](Kahler.md)?
- What is a [spin](spin.md)?
- What is the [Clifford algebra](Clifford%20algebra.md)?
- What is the [Dirac equation](Dirac%20equation.md)


Useful techniques:  [fiber sum](fiber%20sum) and [rational blowdown](rational%20blowdown)