Immersions of punctured 4-manifolds: with applications to Quantum Cellular Automata

TL;DR

The paper develops a detailed immersion theory for punctured closed 4-manifolds, establishing a Postnikov-based obstruction framework that reduces immersion existence to maps between Postnikov 2-types and associated $w_1,w_2$ data, with a shift criterion governing when a $\varphi$-immersion exists. It constructs concrete immersion equivalence classes for manifolds with cyclic and $\mathbb{Z}^4$ fundamental groups, revealing explicit order graphs (e.g., $S^4< M(2)< M(4)< \cdots< \mathbb{CP}^2$) and divisibility-type relations among invariants. The work connects this geometric theory to quantum cellular automata by showing how immersions pull back QCAs, defining a transfer partial order that encodes how topological quantum information can be ported across manifolds, and delivering a robust framework for understanding which QCAs survive under immersion pullback. The appendices provide a detailed 3-manifold analysis (including a rank-2 non-integrable $w_1$) and a complete discussion of the QCA pullback mechanism, wrapping a comprehensive treatment of immersion order and its applications to both topology and quantum information. Overall, the paper advances both the topology of 4-manifolds and the interplay between manifold topology and quantum information processes through a precise, computable invariant-based order structure.

Abstract

Motivated by applications to pulling back quantum cellular automata from one manifold to another, we study the existence of immersions between certain smooth 4-manifolds. We show that they lead to a very interesting partial order on closed 4-manifolds.

Figures (7)

• Figure 1: An immersion of $T^2_*$ into the plane.
• Figure 4: The result is an $\alpha^\prime$ of much smaller range.
• Figure 5:
• Figure 6:
• Figure 7: Arrows are gates and squares commute.

Theorems & Definitions (99)

• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10