Compactification, topology change and surgery theory

TL;DR

The paper explores how topology change, including compactification, can be understood through surgery theory within the causal Lorentzian framework, extending known results to higher dimensions and showing no causal continuity-based selection rule.

Contribution

It demonstrates that topology change in dimensions five and higher can be achieved via causally continuous cobordisms, extending previous 4D results, and highlights the relevance of surgery theory for understanding these processes.

Findings

Topology change in ≥5 dimensions can be causally continuous.

Surgery and handle theory are effective tools for studying topology change.

No causal continuity-based selection rule for compactification.

Abstract

We study the process of compactification as a topology change. It is shown how the mediating spacetime topology, or cobordism, may be simplified through surgery. Within the causal Lorentzian approach to quantum gravity, it is shown that any topology change in dimensions $\geq 5$ may be achieved via a causally continuous cobordism. This extends the known result for 4 dimensions. Therefore, there is no selection rule for compactification at the level of causal continuity. Theorems from surgery theory and handle theory are seen to be very relevant for understanding topology change in higher dimensions. Compactification via parallelisable cobordisms is particularly amenable to study with these tools.