What is...the Poincaré conjecture?

Goal.
Explaining basic concepts of geometric topology in an intuitive way.

This time.
What is...the Poincaré conjecture? Or: Spheres will be spheres.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Disclaimer.
Geometric topology is usually the study of manifolds. This can mean manifold things ;-) So to be precise, this video series is mostly about knots, surfaces, three and four manifolds.

Warning.
There is a historical error in the video and on the slides: As pointed out in the comments, the topological n=4 case is due to Freedman not Smale.

Slides.
http://www.dtubbenhauer.com/youtube.html

Website with exercises.
http://www.dtubbenhauer.com/lecture-g...

Thumbnail.
Picture from    • What is...exotic four space?  

Geometric topology.
https://en.wikipedia.org/wiki/Geometr...
https://en.wikipedia.org/wiki/List_of...
https://en.wikipedia.org/wiki/Manifold
https://en.wikipedia.org/wiki/Knot_(m...)
https://en.wikipedia.org/wiki/Surface...)
https://en.wikipedia.org/wiki/3-manifold

Concepts in geometric topology.
https://en.wikipedia.org/wiki/Knot_in...
https://en.wikipedia.org/wiki/Braid_g...
https://en.wikipedia.org/wiki/Euler_c...
https://en.wikipedia.org/wiki/Mapping...
https://en.wikipedia.org/wiki/Exotic_...
https://en.wikipedia.org/wiki/Poincar...
https://en.wikipedia.org/wiki/Thursto...
https://en.wikipedia.org/wiki/Cobordism
https://en.wikipedia.org/wiki/Unknott...

Applications of geometric topology.
https://physics.stackexchange.com/que...
https://mathoverflow.net/questions/48...
https://math.mit.edu/research/highsch...

The wild world of four manifolds.
https://en.wikipedia.org/wiki/4-manifold
https://en.wikipedia.org/wiki/Categor...
https://arxiv.org/abs/math/0610700
https://web.stanford.edu/~cm5/4D.pdf
https://bookstore.ams.org/gsm-20
https://global.oup.com/academic/produ...
https://en.wikipedia.org/wiki/Four-di...

History of topology.
https://www.elsevier.com/books/histor...
https://en.wikipedia.org/wiki/Analysi...)
https://gallica.bnf.fr/ark:/12148/bpt...
https://www.maths.ed.ac.uk/~v1ranick/...

Poincare conjecture and h-cobordism.
https://en.wikipedia.org/wiki/Poincar...
https://en.wikipedia.org/wiki/General...
https://www.jstor.org/stable/1970239?...
https://en.wikipedia.org/wiki/H-cobor...
https://mathworld.wolfram.com/h-Cobor...

Pictures used.
Picture from https://gallica.bnf.fr/ark:/12148/bpt...
https://encrypted-tbn0.gstatic.com/im...
https://encrypted-tbn0.gstatic.com/im...
https://intuitivescienceblog.files.wo...
Picture from https://www.jstor.org/stable/1970239?...
https://en.wikipedia.org/wiki/Cobordi...
Picture from    • What is...exotic four space?  

Some books I am using (I sometimes steal some pictures from there).
https://www.math.cuhk.edu.hk/course_b...
https://arxiv.org/abs/1610.02592?cont...
https://www.degruyter.com/document/do...
https://bookstore.ams.org/fourman
https://www.degruyter.com/document/do...
https://bookstore.ams.org/surv-55
https://bookstore.ams.org/gsm-20

KnotAtlas.
http://katlas.math.toronto.edu/wiki/T...

SnapPy.
https://snappy.math.uic.edu/

Mathematica.
http://katlas.org/wiki/The_Mathematic...
https://demonstrations.wolfram.com/Ch...

SageMath.
https://doc.sagemath.org/html/en/refe...

#geometrictopology
#topology
#mathematics