Topology of multiple log transforms of 4-manifolds
Selman Akbulut
Michigan State University
MPIM
April 30, 2012
4-dim handlebodies
2-handles
1-handle
simplify
Carving
S
3
4
B
0
handle slide
()
1
0
he lives in 4-manifold !
Figure:
Topology of multiple log
Deforming associatives in G2 manifolds
Selman Akbulut
March 16, 2012
Octonions
Octonions: The division algebra O = H H = R8 is generated by
1, i , j , k , l , li , lj , lk by the multiplication rule:
i 2 = j 2 = k 2 = l 2 = 1 , and :
lk
(li)k = lj
l(ik) =
4-boyutda uzay insaati
Selman Akbulut
September 5, 2011
4-boyutlu uzay (manifold)
R
4
M
R
4
3
R (zaman)
()
4-boyutda uzay insaati
September 5, 2011
2 / 15
4-manifoldun kulplarla tasviri (Morse fonksiyonu)
k-kulp: B 4 = Bk B 4k (k = 0, 1, 2, 3, 4) kenari o
Visualising 4-manifolds
Selman Akbulut
July 30, 2011
4-manifold
R
4
M
R
4
3
R (time)
()
Visualising 4-manifolds
July 30, 2011
2 / 12
4-manifold as a handlebody (Morse function)
A k-handle is just a ball B 4 = Bk B 4k (k = 0, 1, 2, 3, 4) attached
along Bk
Double knot surgeries to S 4 and S 2 S 2
Selman Akbulut
July 15, 2011
4-manifold
R
4
M
R
4
3
R
()
Double knot surgeries to S 4 and S 2 S 2
July 15, 2011
2 / 17
4-manifold as a handlebody (Morse function)
A k-handle is just a ball B 4 = Bk B 4k (k = 0, 1,
Corks, plugs and Stein manifolds
Selman Akbulut
Michigan State University
January 7, 2011
4-manifolds are spaces which are locally R 4
M
R
R
4
()
4
3
Corks, plugs and Stein manifolds
January 7, 2011
2 / 18
We visualize 4-manifolds by handles
B4
3-handles
Knot surgery and Scharlemann manifolds
Selman Akbulut
Michigan State University
January 7, 2011
4-manifolds are spaces that are locally R 4
M
R
R
4
()
4
3
Knot surgery and Scharlemann manifolds
January 7, 2011
2 / 14
Handles of 4-manifolds:
A k-handle is
Dolgachev surface
Selman Akbulut
Michigan State University
April 12, 2010
4-manifolds are spaces that locally look like R 4
M
R
R
4
()
4
3
Dolgachev surface
April 12, 2010
2 / 22
4-Manifolds, as you see it
1-handle
2-handles
S
3-handles
3
4
B
0
1
0
handle
Cappell-Shaneson homotopy 4-spheres are
standard
Selman Akbulut
Michigan State University
April 3, 2010
4-manifolds are spaces that locally look like R 4
M
R
R
4
()
4
3
Cappell-Shaneson homotopy 4-spheres are standard
April 3, 2010
2 / 16
We visualize 4-m