Degree-one maps, surgery and four-manifolds
Siddhartha Gadgil

TL;DR
This paper characterizes degree-one maps between closed, oriented 3-manifolds using surgery on unknotted links and relates their existence to embeddings in specific 4-manifolds, connecting to topological field theories.
Contribution
It provides a surgery-based criterion for degree-one maps and links this to embeddings in 4-manifolds, offering new insights into 3- and 4-dimensional topology.
Findings
Degree-one maps correspond to surgeries on unknotted links.
Existence of degree-one maps is equivalent to certain embeddings in 4-manifolds.
Connects topological invariants to geometric constructions.
Abstract
We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold to a closed, oriented 3-manifold if and only if can be obtained from by surgery about a link in each of whose components is an unknot. We use this to interpret the existence of degree-one maps between closed 3-manifolds in terms of smooth 4-manifolds. More precisely, we show that there is a degree-one map from to if and only if there is a smooth embedding of in W=(N\times I)#_n \bar{\C P^2}#_m {\C P^2}, for some , which separates the boundary components of . This is motivated by the relation to topological field theories, in particular the invariants of Ozsvath and Szabo.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research
