# Diarmuid John CrowleyUniversity of Melbourne | MSD · School of Mathematics and Statistics

Diarmuid John Crowley

PhD, Indiana University 2002

## About

59

Publications

2,502

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

427

Citations

Citations since 2016

Introduction

Diarmuid John Crowley currently works at the School of Mathematics and Statistics, University of Melbourne. Diarmuid does research in Geometry and Topology.

**Skills and Expertise**

## Publications

Publications (59)

We show that for every odd prime $q$, there exists an infinite family $\{M_i\}_{i=1}^{\infty}$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $M_i$ have isometric rank one equivariant intersection pairings and boundary $L(2q, 1) \# (S^1 \times S^2)$, but they are pairwise not homotopy equivalent via an...

For every $k \geq 2$ and $n \geq 2$ , we construct n pairwise homotopically inequivalent simply connected, closed $4k$ -dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension four, we exhibit an analogous phenomenon for spin $^{...

We define a turning of a rank-$2k$ vector bundle $E \to B$ to be a homotopy of bundle automorphisms $\psi_t$ from $\mathbb{Id}_E$, the identity of $E$, to $-\mathbb{Id}_E$, minus the identity, and call a pair $(E, \psi_t)$ a turned bundle. We investigate when vector bundles admit turnings and develop the theory of turnings and their obstructions. I...

We exhibit the first examples of closed 7-dimensional Riemannian manifolds with holonomy G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} that are...

For every $k \geq 2$ we construct infinitely many $4k$-dimensional manifolds that are all stably diffeomorphic but pairwise not homotopy equivalent. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In fact we construct infinitely many such infinite sets. To achieve this we prove a realisation result for appropr...

For every $k \geq 2$ and $n \geq 2$ we construct $n$ pairwise homotopically inequivalent simply-connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension $4$, we exhibit an analogous phenomenon for spin$^{c}$...

We give an elementary topological obstruction for a $(2q{+}1)$-manifold $M$ to admit a contact open book with flexible Weinstein pages: if the torsion subgroup of the $q$-th integral homology group is non-zero, then no such contact open book exists. We achieve this by proving that a symplectomorphism of a flexible Weinstein manifold acts trivially...

For the projective unitary group $PU_n$ with a maximal torus $T_{PU_n}$ and Weyl group $W$, we show that the integral restriction homomorphism \[\rho_{PU_n} \colon H^*(BPU_n;\mathbb{Z})\rightarrow H^*(BT_{PU_n};\mathbb{Z})^W\] to the integral invariants of the Wely group action is onto. We also present several rings naturally isomorphic to $H^*(BT_...

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.

We prove that the derivative map $d \colon \mathrm{Diff}_\partial(D^k) \to \Omega^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that the following homomorphism is non-zero: $$ d_* \colon \pi_5\mathrm{Diff}_\partial(D^{11}) \to \pi_{5}\Omega^{11}SO_{1...

We give necessary and sufficient conditions for a closed orientable 9-manifold M to admit an almost contact structure. The conditions are stated in terms of the Stiefel-Whitney classes of M and other more subtle homotopy invariants of M. By a fundamental result of Borman, Eliashberg and Murphy, M admits an almost contact structure if and only if M...

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures o...

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures o...

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures o...

In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy \(G_2\) metrics on a closed 7-manifold, or to distinguish \(G_{2}\)-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum constructions used to realis...

We prove the "Sullivan Conjecture" on the classification of 4-dimensional complete intersections up to diffeomorphism. Here an $n$-dimensional complete intersection is a smooth complex variety formed by the transverse intersection of $k$ hypersurfaces in $CP^{n+k}$. Previously Kreck and Traving proved the 4-dimensional Sullivan Conjecture when 64 d...

In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy 2 metrics on a closed 7-manifold, or to distinguish 2-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum constructions used to realise 2-manifolds...

We establish upper bounds of the indices of topological Brauer classes over a closed orientable 8-manifolds. In particular, we verify the Topological Period-Index Conjecture (TPIC) for topological Brauer classes over closed orientable 8-manifolds of order not congruent to 2 mod 4. In addition, we provide a counter-example which shows that the TPIC...

The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.

We study the cardinality of the set of manifolds homotopy equivalent to a given manifold M and compare it to the cardinality of the structure set of M.

In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy G_2 metrics on a closed 7-manifold, or to distinguish G_2-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum constructions used to realise G_2-mani...

The Topological Period-Index Conjecture is an hypothesis which relates the period and index of elements of the cohomological Brauer group of a space. It was identified by Antieau and Williams as a topological analogue of the Period-Index Conjecture for function fields. In this paper we show that the Topological Period-Index Conjecture holds and is...

We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. When j=1 we extend the above to certain classes of simply-connected non-spin 5-manifolds. The condition of 2-positive Ricci curvature is...

https://arxiv.org/abs/1612.04776
We work in the smooth category. Let N be a closed connected orientable 4-manifold with torsion free H_1, where H_q:=H_q(N;Z). Our main result is a readily calculable classification of embeddings N→R^7 up to isotopy, with an indeterminancy. Such a classification was only known before for H_1=0 by our earlier work fr...

We construct non-trivial elements of order 2 in the homotopy groups $\pi_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence b...

A Poincar\'e-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 J\"{a}nich presented a Poincar\'{e}-Hopf theorem for line field...

https://arxiv.org/abs/1611.04738
We work in the smooth category. Let N be a closed connected orientable 4-manifold with torsion free H_1, where H_q:=H_q(N;Z). Our main result is a complete readily calculable classification of embeddings N→R^7, up to the equivalence relation generated by isotopy and embedded connected sum with embeddings S^4→R^7. S...

We define the Binachi-Massey tensor on the degree n cohomology with rational
coefficients of a topological space X as a linear map from a subspace of the
fourth tensor power of H^n(X) (determined by the cup product H^n(X) x H^n(X) ->
H^{2n}(X)) to H^{4n-1}(X). If M is a closed (n-1)-connected (4n-1)-manifold
(and n > 1) then its rational homotopy t...

The first and third authors have constructed a defect invariant $\nu(M,\phi)$
in Z/48 for G_2-structures on a closed 7-manifold. We describe the nu-invariant
using $\eta$-invariants and Mathai-Quillen currents on M and show that it can
be refined to an integer-valued invariant $\bar\nu(M,g)$ for G_2-holonomy
metrics. As an example, we determine the...

We exhibit the first examples of closed 7-dimensional Riemannian manifolds
with holonomy G_2 that are homeomorphic but not diffeomorphic. These are also
the first examples of closed Ricci-flat manifolds that are homeomorphic but not
diffeomorphic. The examples are generated by applying the twisted connected sum
construction to Fano 3-folds of Picar...

We give a bordism-theoretic characterization of those closed almost contact $(2q{+ }1)$-manifolds (with $q\geq 2$) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's $h$-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected
almost contact 7-mani...

We present a classification theorem for closed smooth spin 2-connected
7-manifolds M. This builds on the almost-smooth classification from the first
author's thesis. The main additional ingredient is an extension of the
Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether
the spin characteristic class p_M in the fourth integ...

For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected manifold. If $k
\equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then
there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a
Ricci positive metric. This follows from a new description of these manifolds
as the boundaries of explicit plumbings.

We show that if a manifold M admits a contact structure, then so does M × S2. Our proof relies on surgery theory, a theorem of Eliashberg on contact surgery and a theorem of Bourgeois showing that if M admits a contact structure then so does M × T2.

Let X be a closed m-dimensional spin manifold which admits a metric of
positive scalar curvature and let Pos(X) be the space of all such metrics. For
any g in Pos(X), Hitchin used the KO-valued alpha-invariant to define a
homomorphism A_{n-1} from \pi_{n-1}(Pos(X) to KO_{m+n}.
He then showed that A_0 is not 0 if m = 8k or 8k+1 and that A_1 is not 0...

We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new
invariant, the t-invariant, for quaternionic line bundles E over closed
spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in
H^4(M) is torsion. The t-invariant classifies closed smooth oriented
2-connected rational homology 7-spheres up to almost-diffeomorp...

The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL)
manifold with Kervaire invariant 1 and the same homology as the product of two
(2k+1)-dimensional spheres. We show that a finite group of odd order acts
freely on a Kervaire manifold if and only if it acts freely on the
corresponding product of spheres. If the Kervaire mani...

We show that if a manifold M admits a contact structure, then so does M\times
S^2. Our proof relies on surgery theory, a theorem of Eliashberg on contact
surgery and a theorem of Bourgeois showing that if M admits a contact structure
then so does M\times T^2.

We define a Z/48-valued homotopy invariant nu of a G_2-structure on the
tangent bundle of a closed 7-manifold in terms of the signature and Euler
characteristic of a coboundary with a Spin(7)-structure. For manifolds of
holonomy G_2 obtained by the twisted connected sum construction, the associated
torsion-free G_2-structure always has nu = 24. Som...

Fix an integer m and a multi-index p = (p_1, ..., p_r) of integers p_i < m-2.
The set of links of codimension > 2, with multi-index p, E(p, m), is the set of
smooth isotopy classes of smooth embeddings of the disjoint union of the
p_i-spheres into the m-sphere. Haefliger showed that E(p, m) is a finitely
generated abelian group with respect to embe...

Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0. Our main result is the following classification of the set E 7(N) of smooth embeddings N → ℝ7 up to smooth isotopy. Haefliger proved that E7(S4) together with the connected sum operation is a group isomorphic to ℤ12. This group acts on E7(N) by an embedded connected sum. Boéchat an...

The Kreck monoids l 2q+1(ℤ[π]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall surgery obstruction groups, L 2q+1s (ℤ[π]) ⊂ l 2q+1(ℤ[π]). In this paper we identify l 2q+1(ℤ[π]) as the edge set of a directed graph with vertices a set of equivalence classes of quadratic forms...

A functorial semi-norm on singular homology is a collection of semi-norms on
the singular homology groups of spaces such that continuous maps between spaces
induce norm-decreasing maps in homology. Functorial semi-norms can be used to
give constraints on the possible mapping degrees of maps between oriented
manifolds. In this paper, we use informat...

In this note we give examples in every dimension m ≥ 9 of piecewise linearly homeomorphic, closed, connected, smooth m-manifolds which admit two smoothness structures with differing spans, stable spans, and immersion co-dimensions. In dimen-sion 15 the examples include the total spaces of certain 7-sphere bundles over S 8 . The construction of such...

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S
p
× S
q
, for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the...

For a closed topological manifold M with dim (M) >= 5 the topological
structure set S(M) admits an abelian group structure which may be identified
with the algebraic structure group of M as defined by Ranicki. If dim (M) =
2d-1, M is oriented and M is equipped with a map to the classifying space of a
finite group G, then the reduced rho-invariant d...

For \({M_r := \sharp_r(S^p \times S^p),\,p=3, 7}\), we calculate \({\pi_0{\rm Diff}(M_r)/\Theta_{2p+1}}\) and \({\mathcal{E}(M_r)}\), respectively the group of isotopy classes of orientation preserving diffeomorphisms of M
r
modulo isotopy classes with representatives which are the identity outside a 2p-disc and the group of homotopy classes of ori...

We calculate the smooth structure set of $S^p \times S^q$, $S(p, q)$, for $p, q \geq 2$ and $p+q \geq 5$. As a consequence we show that in general $S(4j-1, 4k)$ cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map $F: S(4j, 4k) --> S^{Top}(4j, 4k)...

The monoids l_{2q+1}(Z[\pi]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall simple surgery obstruction groups, L_{2q+1}^s(Z[\pi]) \subset l_{2q+1}(Z[\pi]). In this paper we give exact sequences which completely describe l_{2q+1}(Z[\pi]) as a set and which we use to compute...

Let N be a closed, connected, smooth 4-manifold with H_1(N;Z)=0. Our main
result is the following classification of the set E^7(N) of smooth embeddings
N->R^7 up to smooth isotopy. Haefliger proved that the set E^7(S^4) with the
connected sum operation is a group isomorphic to Z_{12}. This group acts on
E^7(N) by embedded connected sum. Boechat and...

We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller (Ann. of Math. (2) 152 (2000) 331–367) th...

Let P be a closed smooth (4j-2)-connected 8j-manifold. We complete Wilkens' classification of the manifolds P for j = 1,2 and give an alternative proof to Wall's classification of the manifolds for j > 2. The Hopf-invariant-one dimensions (j=1,2) are characteristed by the fact that the quadratic linking functions which classify may be inhomogeneous...

We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller that each of these total spaces admits me...

A systematic consideration of the problem of the reduction and extension of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of the Dixmier-Douady class in various contexts including string structures, U-res bundles and other examples moti...

The Kreck monoids l2q+1(Z(�)) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and gen- eralise the Wall surgery obstruction groups, Ls2q+1(Z(�)) � l2q+1(Z(�)). In this paper we identify l2q+1(Z(�)) as the edge set of a directed graph with vertices a set of equivalence classes of quadratic forms on...

## Projects

Projects (2)

The aim of this project is to understand algebraic invariants such as (generalized) cohomology and Chow rings of the classifying spaces/stacks of the projective linear groups PGL(n).

The goal is to study the following major classical problems of topology.
Knotting Problem: Classify embeddings of a given space into another given space up to isotopy.
Embedding Problem: Find the least dimension m such that given space admits an embedding into m-dimensional Euclidean space R^m.