Torsion Free Homology Research Articles

Existence of Riemannian metrics with positive biorthogonal curvature on simply connected 5-manifolds

Using the recent work of Bettiol, we show that a first-order conformal deformation of Wilking’s metric of almost-positive sectional curvature on \(S^2\times S^3\) yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel–Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes.

On definite lattices bounded by integer surgeries along knots with slice genus at most 2

We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given for the ( 2 , 5 ) (2,5) -torus knot, and analogous results are obtained for integer surgeries on knots of slice genus at most 2. The proofs use input from Yang–Mills instanton gauge theory, Heegaard Floer correction terms, and the topology of singular complex plane curves.

Stable homotopy classification of A n 4 -polyhedra with 2- torsion free homology

In this paper, we study the stable homotopy types of $\mathbf{F}^4_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+4)$-dimensional polyhedra with 2-torsion free homologies. We are able to classify the indecomposable $\mathbf{F}^4_{n(2)}$-polyhedra. The proof relies on the matrix problem technique which was developed in the classification of representaions of algebras and applied to homotopy theory by Baues and Drozd.

The Weinstein Conjecture for Connected Sums

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

A new construction of $6$-manifolds

This paper provides a topological method to construct all simply-connected, spin, smooth $6$-manifolds with torsion-free homology using simply-connected, smooth $4$-manifolds as building blocks. We explicitly determine the invariants that classify these $6$-manifolds from the intersection form and specific homology classes of the $4$-manifold building blocks.

Classification of stable homotopy types with torsion-free homology

We compute the number of indecomposable stable homotopy types with finitely generated torsion free homology of stable dimension k⩾0.

Infinite loop spaces with odd torsion-free homology

It is shown that an infinite loop space with no odd torsion in its integral homology also has no odd torsion in its homotopy. Combined with known results of Steve Wilson, this gives a complete classification; all such spaces are products of the Wilson spaces, which are the building blocks of the spaces in the omega spectrum for BP.

Wilson spaces and stable splittings of BTr

Let Q(X) denote and let BTr denote the classifying space of the r-torus. In [8], Segal showed that Q(BT1) is homotopy equivalent to a product BU × F where BU denotes the classifying space for stable complex vector bundles and F a space with finite homotopy groups. This result has been a very useful one. For example, in [5] it was used to show that up to a stable homotopy equivalence there is only one loop structure on the 3-sphere at each odd prime p. (The subsequent work of Dwyer, Miller, and Wilkerson shows this result is even true unstably, at every prime p.) In [6] it was used to classify, up to homology, the stable self maps of the projective spaces ℂPn and ℍPn. In [5] I asked if a splitting similar to Segal's might exist for Q(BTr) when r≥2. In particular, since the homotopy and homology groups of BU are torsion free it seemed natural to ask if Q(BTr), when r>, could likewise contain a retract with torsion free homology and homotopy groups and whose complement is rationally trivial. The purpose of this note is to show that the answer is no.

Some Calculations In the Unstable Adams-Novikov Spectral Sequence

The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {Er(X)}9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £2term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S2+1. The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E2-term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S2n+1. We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the

Constructing Structurally Stable Diffeomorphisms

One of the central problems of geometric dynamics is to understand the relation of the dynamics of a system to the topology of the manifold on which it occurs-what kinds of dynamics can occur on which manifolds. We are concerned here with this problem as it relates to Smale diffeomorphisms (structurally stable, Axiom A diffeomorphisms with zero dimensional basic sets, see ? 1 for all definitions). This is a large class and includes a representative of every isotopy class. For these diffeomorphisms, the action on a basic set (which is the dynamics in which we are interested) is topologically conjugate to a subshift of finite type and hence specified entirely by a non-negative integer matrix. Our question then becomes: On a given compact manifold M when does there exist in a given homotopy class a diffeomorphism f whose basic sets are the subshifts of finite type corresponding to a set of pre-assigned matrices and occurring with pre-assigned index? (See [10] also.) For the homotopy class of the identity on a large class of manifolds (including all simply-connected manifolds of dimension greater than five with torsion-free homology) we are able to answer this question up to a finite power. That is, we give necessary and sufficient conditions for the existence of a diffeomorphism f isotopic to the identity whose restrictions to its basic sets are the kth powers of certain pre-assigned subshifts of finite type, each occurring with pre-assigned index. These necessary and sufficient conditions are immediately checkable and depend only on the matrices given and the Betti numbers of the manifold. A somewhat more general sufficient condition for the existence f can be given in terms of the existence of a Morse function on M with certain type numbers. Recall that the jt type number of a Morse function is the number of critical points of index j it has. Since the direct sum of matrices corresponds to the disjoint union of the corresponding subshifts of finite type, we can describe all basic sets of a given index (see definition in ? 1) for a Smale diffeomorphism in a single matrix.

On the Classification of $(n-2)$-connected $2n$-manifolds with Torsion Free Homology Groups

On the Classification of $(n-2)$-connected $2n$-manifolds with Torsion Free Homology Groups

Classification of certain 6-manifolds

In (13) Wall classified up to diffeomorphism, PL-homeomorphism, topological homeomorphism, and homotopy equivalence all closed, oriented, 1-connected smooth (or PL) 6-manifolds with torsion-free homology and vanishing second Stiefel-Whitney class. This paper extends these classifications to all closed, oriented, 1-connected topological 6-manifolds with torsion-free homology. The method differs from that of (13) but uses two special cases of Wall's classification.

Smoothings of sphere bundles over spheres in the stable range

The general problem of classifying up to orientation-preserving diffeomorphism those smooth manifolds homeomorphic to a given manifold is probably too complicated to be treated in any effective uniform manner. The first case to be treated was of course the sphere, where everything has been established modulo computation of the Adams spectral sequence. The next case considered was a product of two spheres, where the author and DeSapio independently reduced the classification to homotopy theory with the exception of determining the action of ~P4k + 2 on some (4 k + 1)-dimensional products. It is completely straightforward to extend this classification to the k-sphere bundles associated to (k+l)-plane bundles over S" which have nowhere zero cross sections and satisfy k n which apparently diverge from [1] in some respects. We make two remarks for completeness. A classification of smoothings of bundles with k < n (not necessarily having cross sections) may be derived from [3, w 5]. Finally, throughout this paper the phrases "combinatorially equivalent" and "homeomorphic" are interchangeable by the Hauptvermutung for simply connected closed manifolds with torsion free homology and dimension at least 5.