[Paper Review] Transfer and Chern classes for extraspecial p-groups

David J. Green, Phạm Anh Minh|arXiv (Cornell University)|Jan 1, 1996

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper investigates the cohomology ring of extraspecial p-groups by analyzing the subring generated by Chern classes and transfers, showing it is strictly larger than the Chern subring yet still not the full cohomology ring even modulo nilradicals. A key formula is derived that relates Chern classes to transfer maps, clarifying their interplay in this algebraic structure.

ABSTRACT

In the cohomology ring of an extraspecial p-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers.

Motivation & Objective

• To understand the structure of the cohomology ring of extraspecial p-groups.
• To examine the subring generated by Chern classes and transfers within this ring.
• To determine whether this subring equals the full cohomology ring modulo nilradicals.
• To derive a formula connecting Chern classes and transfer maps in this context.

Proposed method

• Analyzes the cohomology ring of extraspecial p-groups using algebraic topology techniques.
• Identifies and studies the subring generated by Chern classes and transfer maps.
• Applies transfer maps from subgroups to construct elements in the cohomology ring.
• Uses algebraic relations to derive a formula linking Chern classes and transfers.
• Compares the generated subring to the full cohomology ring and to the Chern subring.
• Considers the structure modulo the nilradical to assess completeness of the subring.

Experimental results

Research questions

• RQ1 Is the subring generated by Chern classes and transfers equal to the full cohomology ring of an extraspecial p-group?
• RQ2 How does the subring generated by Chern classes and transfers compare to the Chern subring alone?
• RQ3 Does the subring generated by Chern classes and transfers exhaust the cohomology ring modulo the nilradical?
• RQ4 What algebraic relationship exists between Chern classes and transfer maps in this setting?
• RQ5 Can a general formula be derived that expresses Chern classes in terms of transfers?

Key findings

• The subring generated by Chern classes and transfers is strictly larger than the Chern subring in the cohomology ring of an extraspecial p-group.
• Despite its enlargement, this subring remains strictly smaller than the full cohomology ring, even after quotienting by the nilradical.
• A precise formula is established that relates Chern classes to transfer maps in the cohomology of extraspecial p-groups.
• The results demonstrate a non-trivial interaction between characteristic classes and transfer maps in this class of groups.
• The cohomology ring structure is shown to be more complex than previously suggested by Chern class theory alone.