[论文解读] Topological classification of certain nonorientable 4-manifolds with cyclic fundamental group of order 2 mod 4

Rafael Torres|arXiv (Cornell University)|Jan 8, 2026

Geometric and Algebraic Topology被引用 0

一句话总结

Extends Hambleton-Kreck-Teichner’s classification of closed topological nonorientable 4-manifolds with π1 of order 2 to cyclic groups of order 2p (p odd), using a cut-and-paste construction and surgery theory to realize and compare manifolds via invariants such as χ, w1^4, KS, and arf, with partial results toward a complete classification.

ABSTRACT

We show that the classification up to homeomorphism of closed topological nonorientable 4-manifolds with fundamental group of order 2 due to Hambleton-Kreck-Teichner can be used to classify a large set of such 4-manifolds with cyclic fundamental group of order 2p for every odd $p > 1$. This is done through a simple cut-and-paste construction, and classical and modified surgery theory are used only through results of Hambleton-Kreck-Teichner and Khan. It is plausible that this set comprises all closed topological nonorientable 4-manifolds with $π_1 = \Z/2p$. We collect several interesting questions whose answers would guarantee a complete classification.

研究动机与目标

• Extend the Hambleton-Kreck-Teichner classification from π1 of order 2 to π1 = Z/2p for odd p>1.
• Develop a cut-and-paste construction to generate a large collection of such 4-manifolds while preserving key invariants.
• Analyze invariants under construction to relate homeomorphism types before and after the construction.
• Provide partial answers toward a complete classification and identify open questions and conjectures.

提出的方法

• Use Construction A to create X2p by gluing X u(α) to Np along boundary, yielding π1(X2p) = Z/2p.
• Employ invariant analysis (χ, w1^4, KS, arf) to show these are unchanged by Construction A.
• Utilize TopPin^c and Pin^c bordism theories to relate arf and η invariants under primitive structures.
• Apply results of Hambleton-Kreck-Teichner and Khan to compare homeomorphism types via stabilized equivalence.
• Construct a collection of manifolds (Collection B) realizing homeomorphism classes corresponding to initial data leading to X2p.
• Leverage Debray’s stable classification and Khan’s cancellation to address stable vs. actual homeomorphism classifications.

实验结果

研究问题

• RQ1 Does Collection B exhaust all homeomorphism classes of closed nonorientable 4-manifolds with finite cyclic π1 of order 2p (p odd)?
• RQ2 Can the set {χ, w1^4, KS, arf} (and η in the smooth case) classify these manifolds up to homeomorphism?
• RQ3 Can stable homeomorphism classification be upgraded to an actual homeomorphism classification for these manifolds?
• RQ4 How does the cancellation problem for S^2 × S^2 factors affect a complete classification (as in Khan’s results)?

主要发现

• There is a 1-to-1 correspondence between homeomorphism classes in Collection B and the initial data used in Construction A.
• The invariants {χ, w1^4, KS, arf} are preserved by Construction A.
• When smooth structures exist, η-invariants provide additional refinement parallel to arf in distinguishing classes.
• A complete stable classification with 20 classes is achieved via Debray’s results and Kreck’s modified surgery theory, refining to explicit realizations by specific manifolds (e.g., A2p,1, R2p,1, B2p,1 and their variants).
• Theorem F shows that M2p # 2(S^2 × S^2) is homeomorphic to exactly one manifold in Collection B, aligning stabilized classifications with the constructed list.