Conjugation spaces are spaces with an involution such that the fixed point set of the involution has Z 2 -cohomology ring isomorphic to the Z 2 -cohomology of the space itself, with the difference that all degrees are divided by two (e.g. CP n with the complex conjugation has RP n as fixed point set). One also requires that a certain conjugation equation is fulfilled. We give a new characterisation of conjugation spaces and apply it to the following realization problem: given M, a closed orientable 3-manifold, does there exist a simply connected 6-manifold X and a conjugation on X with fixed point set M? We give an affirmative answer. Mathematics Subject Classification (2000)57R91 · 55M35 · 55N91

In this article, we show the existence of conjugations on many simply-connected spin 6-manifolds with free integral cohomology. In a certain class the only condition on X 6 to admit a conjugation with fixed point set M 3 is the obvious one: the existence of a degree-halving ring isomorphism between the Z 2 -cohomologies of X and M .

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold (M, ω). Assume that the fixed point set M S 1 has exactly two components, X and Y , and that dim(X) + dim(Y ) + 2 = dim(M ). We first show that X, Y and M are simply connected. Then we show that, up to S 1 -equivariant diffeomorphism, there are finitely many such manifolds in each dimension. Moreover, we show that in low dimensions, the manifold is unique in a certain category. We use techniques from both areas of symplectic geometry and geometric topology. Mathematics Subject Classification 53D05, 57R65, 53D20, 55Q05, 57R80 (primary). Theorem 1. Under Assumption 1.2, the manifolds M, X and Y are all 1-connected.Lemma 3.2. Let X be a closed orientable topological n-manifold and π : X → X the universal covering map. If π 1 (X) is infinite, then H n ( X; Z) ⊗ Q = 0. If π 1 (X) is finite, then π * : H n (X; Z) → H n ( X; Z) is a degree ±|π 1 (X)| map.Proof of Lemma 3.1. Since P is simply connected, the map f factors through the universal covering map π : X → X, giving us a commutative diagram:H n ( X; Z) f * v v m m m m m m m m m m m m Z ∼ = H n (P ; Z) H n (X; Z) ∼ = Z π *

We compute the completed T M F 0 (3)-cohomology of the 7connected cover BString of BO. We use cubical structures on line bundles over elliptic curves to construct an explicit class which together with the Pontryagin classes freely generates the cohomology ring. Introduction and statement of resultsCharacteristic numbers play an important role in the determination of the structure of cobordism rings. For unoriented, oriented and spin manifolds, the cobordism rings were calculated in the 50s and 60s by Thom [Tho54], Novikov [Nov62] and Anderson, Brown and Peterson [ABP67] with the help of Stiefel-Whitney, HZand KO-Pontryagin numbers. However, it is known that for manifolds with lifts of the tangential structure to the 7-connected cover BString of BO these numbers do not determine the bordism classes.Locally at the prime 2, the Thom spectrum M Spin splits into summands of connected covers of KO and an Eilenberg-MacLane part [ABP67, Theorem 2.2]. A similar splitting is conjectured for M String where KO is replaced by suitable versions of the spectrum T M F : the Witten orientation provides a surjection of the string bordism ring to the ring of topological modular forms [Hop02, Theorem 6.25] and there is evidence [MR09, Section 7] that another summand of M String is provided by the 16-connected cover of T M F 0 (3). In order to provide maps to this possible summand one has to study T M F 0 (3)-characteristic classes for string manifolds which is the subject of this work.In [Lau] the T M F 1 (3)-cohomology rings of BSpin and BString were computed. It turned out [Lau, Theorem 1.2] that the T M F 1 (3)-cohomology ring of BSpin is freely generated by certain classes p i . These deserve the name Pontryagin classes since they share the same properties as the HZ-and KO-Pontryagin classes. In the case of BString, there is another class r coming up which together with the Pontryagin classes freely generates the T M F 1 (3)-cohomology ring when localized at K(2) for the prime 2.The theory T M F 1 (3) is a complex orientable theory. Its formal group is the completion of the universal elliptic curve with Γ 1 (3)-structures. Its relation to T M F 0 (3) is analogous to the relation between complex and real K-theory: a Γ 1 (3)structure is a choice of point of exact order 3 on an elliptic curve. A Γ 0 (3)-structure is the choice of subgroup scheme of the form Z/3 of the points of order 3. Given such a subgroup scheme there are exactly two choices of points of exact order 3 and they differ by a sign. Hence the corresponding cohomology theory T M F 0 (3) is the "real" version of the complex theory T M F 1 (3). It can be obtained by taking Date: June 9, 2018.