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In this paper we prove (integral) homological stability for the sequences of spaces C+n (M,X). These are the spaces of configurations of n points in a connected manifold of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an orientation -- an ordering of the points up to even permutations. They are double covers of the unordered configuration spaces Cn(M,X), and indeed to prove our result we adapt methods from a paper of Randal- Williams, which proves homological stability in the unordered case. Interestingly the oriented configuration spaces stabilise more slowly than the unordered ones: the stability slope we obtain is ⅓, compared to ½ in the unordered case (and these are the best possible slopes in their respective cases). This result can also be interpreted as homological stability for the unordered configuration spaces with certain twisted Z ⊕ Z-coefficients. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5. [ABSTRACT FROM AUTHOR]

For each integer q ≥ 0, there is a cohomology theory A1 such that the zero cohomology group A10 (N) of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n + q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q + 1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q - 1)/2 cobordism groups of framed immersions. The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds. [ABSTRACT FROM AUTHOR]

In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of $X$, where $X$ is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit. [ABSTRACT FROM AUTHOR]

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions. [ABSTRACT FROM AUTHOR]

We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group G of a closed orientable manifold on a zero-dimensional compact space Z, this invariant corresponds to the space of finite Borel measures on Z which are invariant under the action of G. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures. [ABSTRACT FROM AUTHOR]

We calculate the intersection ring of 3-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary 3-manifolds from being homology cobordant to certain graph manifolds. [ABSTRACT FROM AUTHOR]

A conjecture of Vakil and Wood (2015) states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range. [ABSTRACT FROM AUTHOR]

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds with respect to forming the connected sum with the product of spheres Sp x Sq, for p < q < 2p-2. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension 2n > 4 with respect to forming connected sums with Sn x Sn. [ABSTRACT FROM AUTHOR]

The main theorem of this article provides sufficient conditions for a degree d finite cover M' of a hyperbolic 3-manifold M to be a surface bundle. Let F be an embedded, closed and orientable surface of genus g, close to a minimal surface in the cover M', splitting M' into a disjoint union of q handlebodies and compression bodies. We show that there exists a fiber in the complement of F provided that d, q and g satisfy some inequality involving an explicit constant k depending only on the volume and the injectivity radius of M. In particular, this theorem applies to a Heegaard splitting of a finite covering M', giving an explicit lower bound for the genus of a strongly irreducible Heegaard splitting of M'. Applying the main theorem to the setting of a circular decomposition associated to a non-trivial homology class of M gives sufficient conditions for this homology class to correspond to a fibration over the circle. Similar methods also lead to a sufficient condition for an incompressible embedded surface in M to be a fiber. [ABSTRACT FROM AUTHOR]

We generalise the Kreck-Stolz invariants s2 and s3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spinmanifolds M of dimension 4k-1 with H³(M;Q) = 0 such that c2(E) ? H4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-spheres up to almost-diffeomorphism, that is, diffeomorphism up to a connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The t-invariant also gives information about quaternionic line bundles over a fixed manifold, and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over HPk. The t-invariant for S4k-1 is closely related to the Adams e-invariant on the (4k - 5)-stem. [ABSTRACT FROM AUTHOR]

We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain "Relative Morse Inequalities" relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman's Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3 and for each k ≥ 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d - 1)-cells. (This solves a problem by Chari.) (2) For fixed d and k, there are exponentially many combinatorial types of simplicial d-manifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d - 1)- cells. (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.) (3) The barycentric subdivision of any simplicial constructible d-ball is collapsible. (This "almost" solves a problem by Hachimori.) (4) Every constructible ball collapses onto its boundary minus a facet. (This improves a result by the author and Ziegler.) (5) Any 3-ball with a knotted spanning edge cannot collapse onto its boundary minus a facet. (This strengthens a classical result by Bing and a recent result by the author and Ziegler.) [ABSTRACT FROM AUTHOR]

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincaré dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-selfdual SO(3) connections on a closed four-manifold. This should be a useful tool in comparing gauge theoretic invariants of smooth four-manifolds. [ABSTRACT FROM AUTHOR]

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes the non-simply connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology. [ABSTRACT FROM AUTHOR]