The D(2) problem is a fundamental problem in low dimensional topology. In broad terms, it asks when a three-dimensional space can be continuously deformed into a two-dimensional space without changing the essential algebraic properties of the spaces involved.The problem is parametrized by the fundamental group of the spaces involved; that is, each group G has its own D(2) problem whose difficulty varies considerably with the individual nature of G.This book solves the D(2) problem for a large, possibly infinite, number of finite metacyclic groups G(p, q). Prior to this the author had solved the D(2) problem for the groups G(p, 2). However, for q > 2, the only previously known solutions were for the groups G(7, 3), G(5, 4) and G(7, 6), all done by difficult direct calculation by two of the author's students, Jonathan Remez (2011) and Jason Vittis (2019).The method employed is heavily algebraic and involves precise analysis of the integral representation theory of G(p, q). Some noteworthy features are a new cancellation theory of modules (Chapters 10 and 11) and a simplified treatment (Chapters 5 and 12) of the author's theory of Swan homomorphisms.
Author(s): Francis E. A. Johnson
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 371
City: Singapore
Contents
Preface
Introduction
The groups G(p, q)
The derived module category
Stable modules and syzygies
The Swan homomorphism
Modules over metacyclic groups
Some structural theorems
Identifying syzygies via diagonal resolutions
The D(2)-property
Chapter One. Projective modules and class groups
§1: Exact sequences and projective modules:
§2: The Grothendieck group of a ring:
§3: The reduced Grothendieck group:
§4: Projective modules over R[G]:
§5: Steinitz’ Theorem:
§6: Artinian rings are weakly Euclidean:
§8: The Milnor exact sequence:
§9: Projective modules over Z[Cp]:
Chapter Two. Homological algebra
§10: Cochain complexes and cohomology:
§11: The cohomology theory of modules:
§12: The exact sequences in cohomology:
§13: Module extensions:
§14: The group structure on Ext1:
§15: The cohomological interpretation of Ext1:
§16: The exact sequences of Ext1:
§17: Example, the cyclic group of order m:
Chapter Three. The derived module category
§18: The derived module category:
§19: Stable equivalence and projective equivalence:
§20: Syzygies and generalized syzygies:
§21: The corepresentation theorem for Ext1:
§22: De-stabilization:
§23: The dual to Schanuel’s Lemma:
§24: Endomorphism rings:
Chapter Four. Extension and restriction of scalars
§25: Extension and restriction of scalars:
§26: Transversals and cocycles:
§27: Wreath products and group extensions:
§28: Induced representations:
§29: Lattices and representations:
§30: Induced modules:
§31: Duality:
§32: The Eckmann-Shapiro Theorem:
§33: Syzygies and lattices:
§34: The Eckmann-Shapiro relations in cohomology:
§35: Frobenius reciprocity:
Chapter Five. Swan homomorphisms
§36: Swan’s projectivity criterion:
§37: Tame modules:
§38: The Swan homomorphism:
§39: Invariance properties of the Swan homomorphism:
§40: The relation between consecutive Swan homomorphisms:
§41: The dual Swan mapping:
§42: The stability group of a lattice:
§43: A criterion for monogenicity:
§44: Swan homomorphisms in their original context:
§45: The Swan homomorphism for cyclic groups:
§46: Cancellation over Z[Cm]:
Chapter Six. Modules over quasi-triangular algebras
§47: Modules over Tq(F):
§48: Stable classification of projective modules over Tq(A, I)
§49: A lifting theorem:
§50: Modules over Mq(A):
§51: Classification of projective modules over Tq(A, π):
§52: Properties of the row modules R(i):
§53: Duality properties of the modules R(i):
Chapter Seven. A fibre product decomposition
§54: A fibre product decomposition:
§55: The discriminant of an associative algebra:
§56: The discriminant of a cyclic algebra:
§57: The discriminant of a quasi-triangular algebra:
§58: A quasi-triangular representation of G(p, q):
§59: The isomorphism Cq(I∗, θ)∼=Tq(A, π):
§60: A worked example:
§61: Lifting units to cyclotomic rings:
§62: Liftable subgroups of F∗p×· · ·×F∗p/q
§63: p-adic analogues:
Chapter Eight. Galois modules
§64: Galois modules:
§65: The Galois action of y:
§66: R(1) and R(q) as Galois modules:
§67: The theorem of Auslander-Rim:
§68: Galois module description of the row modules:
Chapter Nine. The sequencing theorem
§69: Basic identities:
§70: Cohomological calculations:
§71: Decomposing the augmentation ideal of Λ:
§72: The basic sequence:
§73: The derived sequences:
§74: Computing the sequencing permutation:
§75: An element of D2k+1(Z):
Chapter Ten. A cancellation theorem for extensions
§76: K-Q-modules:
§77: A strong cancellation semigroup:
§78: K-Q-modules for metacyclic groups:
§79: A cancellation theorem for extensions:
Chapter Eleven. Cancellation of quasi-Swan modules
§80: The EndDer(K)-module structure on Ext1(Q,K)
§81: Nondegenerate modules of type R-Z[Cq]:
§82: Nondegenerate modules of type R-Iq :
§83: A reduction theorem:
§84: An isomorphism theorem:
§85: Rearranging the extension parameters:
§86: A stability theorem:
§87: Straightness of degenerate modules:
Chapter Twelve. Swan homomorphisms for metacyclic groups
§88: Tq(A, π) is full:
§89: A model for STq :
§90: The identity Im(SR(k)) = Im(STq ):
§91: A simplification:
§92: The first modification:
§93: The second modification:
§94: Fullness of R(k):
§95: Changing rings from Z[Cq] to Z[G(p, q)]:
§97: Fullness of R(k) ⊕ [y − 1):
Chapter Thirteen. An obstruction to monogenicity
§98: Obstructions to strong monogenicity:
§99: The unit group of Z[y]/(y6 − 1):
§100: The unit group of F2[C6]:
§101: Projective modules over Z[C12]:
§102: R(1) is not strongly monogenic over Z[G(13, 12)]:
Chapter Fourteen. The D(2) property
§103: Proof of Theorem IV:
§104: Proof of Theorem V:
§105: Proof of Theorems VI, VII and VIII:
§106: The hypotheses K0(Z[Cq]) = 0 and Inj (p, q):
Appendix A: Examples
§A1: The dihedral groups D2p = G(p, 2):
§A2: The groups G(p, 3):
§A3: The groups G(p, 4):
§A4: The groups G(p, 6):
Appendix B: Class field theory and condition Inj(p, q)
§B1: Functorial decomposition of class groups:
§B3: Kida’s Lemma and its consequences:
Appendix C: A sufficient condition for the D(2) property
§C1: A sufficient condition:
§C2: R(2) =⇒ D(2):
§C3: RF(J) =⇒ R(2, J).
§C4: Proof of the Sufficient Condition:
References
Index
