MA5209 Algebraic Topology Wayne Lawton Department of Mathematics National University of Singapore S , Lecture 7. Cohomology Rings (10, 13 November 2009)

Simplicial Cohomology Definition Letbe a simlicial complex, let coboundary operator be an abelian group, and letbe a nonnegative dimensional cochains integer, and define groups: dimensional cocycles dimensional coboundaries dim. cohomology group

Simplicial Cohomology A convenient basis for is then given by simplices A convenient dual basis for where and is the set of oriented Then where ifis an oriented face ofelse

Cohomology of the Torus all 27 edges oriented all 18 faces oriented counterclockwise GroupGenerating Cocycles

Cohomology of the Klein Bottle vertical edges up GroupGenerating Cocycles all 18 faces oriented counterclockwise down except right edges other edgesor Remarksince but

Universal Coefficient Theorem for Cohomology Theorem (UCTC) Letbe a fee chain complex (considered as amodule) and be an abelian group. Then there exists a split exact sequence so Proof p. 320 in Munkres’ Elements Alg. Topology Remark Ext contravariant in 1 st, covariant in 2 nd arg. Proof p in Munkres’ Elem. of Alg. Topology Example Ifis simplicial chain complex of a Klein bottle thenhence

Cup Product on Cochains Fix a commutative (abelian) ring and for all integers with identity Let to be the be a simplicial complex whose vertices are define the cup product ordered by an order denoted by < bilinear function that satisfies where

Cup Product Properties satisfies Exercise Show that the boundary operator and hence makes Corollary The cup product induces a cup product a graded ring. Anticommutativity Exercise Show thatis the mult. identity.

Cup Product for the Torus

Cup Product for the Klein Bottle

Künneth Theorems Let X and Y be topological spaces. Then there exists a split exact sequence and if eachis finitely generateda SES Corollary Proof hence

Ring Structures on Theorem Proof By DeRham’s Theorem is generated as a ring by the 1-forms and these satisfywhere generate and Hence generate generates generate etc. Theorem(polynomial ring). Proof Follows from Poincare Duality for manifolds, Corollary 68.4 on p. 401 in Munkres’ Elem. Alg. Top.