Conjugation spaces and 4-manifolds

Abstract

Abstract. We show that 4-dimensional conjugation manifolds are all obtained from branched 2-fold coverings of knotted surfaces in Z 2 -homology 4-spheres.

Involutions onS6with 3–dimensional fixed point set

“…4.1 in [HHP05], shows that S 2n is a conjugation space. (A slightly different proof is given in [HH09].) q.e.d.…”

“…Recently, a class of involutions called conjugations was defined in [HHP05] and various aspects of conjugations were studied in [FP05,Olb07,HH08,HH09]. Conjugations τ on topological spaces X have the property that the fixed point set has Z 2 -cohomology ring isomorphic to the Z 2 -cohomology ring of X, with the slight difference that all degrees are divided by two.…”

“…In dimension 4, work of Gordon and Sumners shows that there are infinitely many non-equivalent conjugations on S 4 . Hambleton and Hausmann recently reduced the study of such involutions to a non-equivariant fourdimensional knot theory question [HH09]. Knot theory of k-spheres in S 2k is easier for k > 2, so that the study of conjugations gets a different flavor for k > 2.…”

Cobordism groups of simple branched coverings

“…Another example is the quotient map CP 2 → S 4 of the action of Z 2 on CP 2 by conjugation. If the smooth structure of the quotient space (homeomorphic to S 4 ) is chosen appropriately, this map is a branched covering with singular submanifold RP 2 (see Hambleton and Hausmann [8]).…”

Conjugations on 6-manifolds with free integral cohomology

“…In dimension 2, only the sphere admits conjugations. Dimension 4 is special since the quotient space of the involution can be given a smooth structure [HH09]. Hambleton and Hausmann prove that equivariant diffeomorphism classes of oriented connected conjugation 4-manifolds correspond to diffeomorphism classes of pairs (X, Σ), where X is an oriented 4-dimensional Z 2 -homology sphere and Σ is a closed connected subsurface of X.…”