---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "Poincare conjectures"
aliases: [generalized Poincare conjecture, generalized Poincare, "Poincaré", "Poincaré conjecture", "Generalized Poincare Conjectures", smooth Poincare]
---

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

---
- Tags: 
	- #open/conjectures 
- Refs:
	- #todo/add-references
- Links:
	- [smooth structures](smooth%20structures.md)
	- [homology sphere](homology%20sphere.md)
	- [h-cobordism](Unsorted/h-cobordism.md)

---

# Generalized Poincare Conjecture

![](attachments/Pasted%20image%2020220422102330.png)

![](attachments/Pasted%20image%2020220422101003.png)
![](attachments/Pasted%20image%2020220422101104.png)

![](attachments/Pasted%20image%2020220422093408.png)
![](attachments/Pasted%20image%2020220422094057.png)

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# History

Poincaré, *Analysis Situs* papers in 1895. 
Coined "homeomorphism", defined homology, gave rigorous definition of homotopy, established "method of invariants" and essentially kicked off algebraic topology.

# Generalized Topological Poincaré Conjecture

When is a homotopy sphere also a topological sphere? 

When does $\pi_* X \isoas{\Grp} \pi_* S^n \implies X \isoas{\Top} S^n$?

- $n=1$: True. Trivial
- $n=2$: True. Proved by Poincaré, classical
- $n=3$: True. Perelman (2006) using Ricci flow + surgery
- $n=4$: True. Freedman (1982), Fields medal!
- $n=5$: True. Zeeman (1961)
- $n=6$: True. Stalling (1962)
- $n\geq 7$: True. Smale (1961) using h-cobordism theorem, uses handle decomposition + Morse functions

# Smooth Poincaré Conjecture

When is a homotopy sphere a *smooth* sphere?

- $n=1$: True. Trivial
- $n=2$: True. Proved by Poincaré, classical
- $n=3$: True.  (Top = PL = Smooth)
- $n=4$: **Open**
- $n=5$: Zeeman (1961)
- $n=6$: Stalling (1962)
- $n\geq 7$: False in general (Milnor and Kervaire, 1963), Exotic $S^n$, 28 smooth structures on $S^7$
  
**Remarks**:

- It is unknown whether or not $\BB^4 $ admits an exotic smooth structure. If not, the smooth 4-dimensional Poincaré conjecture would have an affirmative answer. 

- Current line of attack: [Gluck twists](Gluck twists) on on $S^4$. 
  	Yield homeomorphic spheres, *suspected* not to be diffeomorphic, but no known invariants can distinguish smooth structures on $S^4$.

# Proofs

![](attachments/Pasted%20image%2020220425091702.png)