Date: Friday, November 20th

Time:  2:00 PM EST

Room: Skiles 006

Title: Exotic phenomena in noncompact four-manifolds.

Committee: John Etnyre (advisor), Jennifer Hom, Johnathan Zung, Robert Gompf (UT Austin), Akram Alishahi (UGA).

Abstract:

We present several results on exotic open 4-manifolds and knotted surfaces in 4-manifolds. These include obstructive, constructive, and qualitative results. 

First, we discuss obstructions to diffeomorphism using end Floer homology (a direct limit of Heegaard Floer homology), and use this to study exotic R4's. This yields the first completely constructive infinite family of exotic R4's. We find more new phenomena about slice R4's, including the first example of distinct slice R4's made using the same slice disk complement and same Casson handle. As another application, we recover a result of Bizaca-Etnyre on smooth structures of four-manifolds with collarable ends. 

Next, we give a general construction of exotic open 2-handles. We use these to find new exotic smooth structures on open 4-manifolds, which have desirable properties. As an application, we construct examples of knotted proper surfaces in R4 which are topologically standard but smoothly distinct, and remain smoothly distinct after end-sum with exotic planes, making progress on a question of Gompf. As more applications of the 2-handles, we construct a family of topologically slice links, generalizing a theorem of Cha-Powell, and prove all open infinite towers in the sense of disk embedding theory are topologically standard. 

Finally, we give the simplest known decomposition of an exotic R4 into two open sets: two thickened circles, glued along the standard thickened wedge of two circles. This makes progress on a question of Kirby, asking whether any exotic R4 is the union of two R4's. Our proof is obtained by studying cuts in Casson-Gordon fibered slice disk complements, and partially attached Casson handles.

Zoom:

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Meeting ID: 952 3463 8837 

Passcode: 623683