Bordism Theory Research Articles

Wess-Zumino-Witten terms of Sp QCD by bordism theory

We investigate the four-dimensional Wess-Zumino-Witten (WZW) terms within the framework of Sp quantum chromodynamics (QCD) using invertible field theory through bordism theory. We present a novel approach aimed at circumventing both perturbative and non-perturbative gauge anomalies on spacetime manifolds endowed with spin structures. We study both ungauged and gauged WZW terms including the problems of the topological consistency of gauged WZW terms.

$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory

We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $\partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU \to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.

$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory

Изучены $SU$-линейные операции в комплексных кобордизмах и доказано, что все они порождаются известными геометрическими операциями $\partial_i$. Для теории $c_1$-сферических бордизмов $W$ описаны все $SU$-линейные умножения на $W$ и проекторы $MU \to W$. Кроме того, исследованы комплексные ориентации на $W$ и соответствующие им формальные группы $F_W$. Связь между формальными группами $F_W$ и кольцом коэффициентов $W_*$ теории $W$ изучалась В. М. Бухштабером в 1972 г. В качестве обобщения этих результатов доказано, что для любых $SU$-линейного умножения и ориентации на $W$ коэффициенты соответствующей формальной группы $F_W$ не порождают все кольцо $W_*$, в отличие от случая комплексных кобордизмов. Библиография: 18 наименований.

Cofreeness in real bordism theory and the segal conjecture

We prove that the genuineC2nC_{2^n}-spectrumNC2C2nMURN_{C_{2}}^{C_{2^n}}MU_{\mathbb {R}}is cofree, for allnn. Our proof is a formal argument using chromatic hypercubes and the Slice Theorem of Hill, Hopkins, and Ravenel. We show that this gives a new proof of the Segal Conjecture forC2C_2, independent of Lin’s theorem.

Models of Lubin–Tate spectra via Real bordism theory

We study certain formal group laws equipped with an action of the cyclic group of order a power of 2. We construct C2n-equivariant Real oriented models of Lubin–Tate spectra Eh at heights h=2n−1m and give explicit formulas of the C2n-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory MUR, and our work examines the height of the formal group laws of the Hill–Hopkins–Ravenel norms of MUR.

Positive scalar curvature metrics via end-periodic manifolds

We obtain two types of results on positive scalar curvature metrics for compact spin manifolds that are even dimensional. The first type of result are obstructions to the existence of positive scalar curvature metrics on such manifolds, expressed in terms of end-periodic eta invariants that were defined by Mrowka-Ruberman-Saveliev (MRS). These results are the even dimensional analogs of the results by Higson-Roe. The second type of result studies the number of path components of the space of positive scalar curvature metrics modulo diffeomorphism for compact spin manifolds that are even dimensional, whenever this space is non-empty. These extend and refine certain results in Botvinnik-Gilkey and also MRS. End-periodic analogs of K-homology and bordism theory are defined and are utilised to prove many of our results.

On the Uniformly Proper Classification of Open Manifolds

We give a brief account on the uniformly proper classification of open manifolds, i.e., the classification under bounded, uniformly proper maps. The small category of diffeomorphism classes of open n-manifolds has uncountably many homotopy types, n ≥ 2. Our main approach consists in splitting this set into generalized components and then trying to classify these components and thereafter the elements inside a component. To define these components, we introduce Gromov–Hausdorff and Lipschitz metrizable uniform structures and corresponding GH- and L-cohomologies. The GH-components are particularly appropriate to introduce geometric bordism theory for open manifolds, and the L-components are appropriate to establish surgery. We present independent generators for the bordism groups. The fundamental contributions of Farrell, Wagoner, Siebenmann, Maumary, and Taylor play a decisive role. In our approach, we suppose the manifolds to be endowed with a metric of bounded geometry and restrict ourselves to bounded uniformly proper morphisms. Finally, we discuss the question under which conditions bounded geometry and uniform properness are preserved by surgery, and sketch some proper surgery groups.

Algebraic realization of actions of some finite groups

Let G be $$A_5$$ , $$A_4$$ , or a finite group with cyclic Sylow 2 subgroup. We show that every closed smooth G manifold M has a strongly algebraic model. This means, there exist a nonsingular real algebraic G variety X which is equivariantly diffeomorphic to M and all G vector bundles over X are strongly algebraic. Making use of improved blow-up techniques and the literature on equivariant bordism theory, we are extending older algebraic realization results.

Derived induction and restriction theory

Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for $G$-equivariant $E$-homology and cohomology, and generalizations of Quillen's $\mathcal{F}_p$-isomorphism theorem when $E$ is a homotopy commutative $G$-ring spectrum. We show that the subcategory $\mathscr{F}^{\mathrm{Nil}}$ contains many $G$-spectra of interest for relatively small families $\mathscr{F}$. These include $G$-equivariant real and complex $K$-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any $L_n$-local spectrum, the classical bordism theories, connective real $K$-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.

Filtered finiteness of the image of the unstable Hurewicz homomorphism with applications to bordism of immersions

AbstractAfter recent work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem [M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. Math. (2), 184 (2016), 1–262], as well as Adams’ solution of the Hopf invariant one problem [J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (2), 72 (1960), 20–104], an immediate consequence of Curtis conjecture is that the set of spherical classes in H∗Q0S0 is finite. Similarly, Eccles conjecture, when specialized to X=Sn with n> 0, together with Adams’ Hopf invariant one theorem, implies that the set of spherical classes in H∗QSn is finite. We prove a filtered version of the above finiteness properties. We show that if X is an arbitrary CW-complex of finite type such that for some n, HiX≃0 for any i>n, then the image of the composition π∗ΩlΣl+2X→π∗QΣ2X→H∗QΣ2X is finite; the finiteness remains valid if we formally replace X with S−1. As an application, we provide a lower bound on the dimension of the sphere of origin on the potential classes of π∗QSn which are detected by homology. We derive a filtered finiteness property for the image of certain transfer maps ΣdimgBG+→QS0 in homology. As an application to bordism theory, we show that for any codimension k framed immersion f:M↬ℝn+k which extends to an embedding M→ℝd×ℝn+k, if n is very large with respect to d and k then the manifold M as well as its self-intersection manifolds are boundaries. Some results of this paper extend results of Hadi [Spherical classes in some finite loop spaces of spheres. Topol. Appl., 224 (2017), 1–18] and offer corrections to some minor computational mistakes, hence providing corrected upper bounds on the dimension of spherical classes H∗ΩlSn+l. All of our results are obtained at the prime p = 2.

Hurewicz images of real bordism theory and real Johnson–Wilson theories

We show that the Hopf elements, the Kervaire classes, and the κ¯-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the C2-fixed points of the Real bordism spectrum. A subset of these families is detected by the C2-fixed points of Real Johnson–Wilson theory ER(n), depending on n. In the proof, we establish an isomorphism between the slice spectral sequence and the C2-equivariant May spectral sequence of BPR.

Analysis, Geometry and Topology of Positive Scalar Curvature Metrics

Riemannian manifolds with positive scalar curvature play an important role in mathematics and general relativity. Obstruction and existence results are connected to index theory, bordism theory and homotopy theory, using methods from partial differential equations and functional analysis. The workshop led to a lively interaction between mathematicians working in these areas.

Differential K-homology and explicit isomorphisms between $\mathbb{R/Z}$-K-homologies

In this paper, we construct an explicit isomorphism between Deeley $\mathbb{R}/\mathbb{Z}$-K-homology and flat K-homology. We also describe $\mathbb{R}/\mathbb{Z}$-K-homology out of $\mathbb{Z}/k\mathbb{Z}$-bordism theories.

Stratified and unstratified bordism of pseudomanifolds

We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples of pseudomanifold bordism theories, such as bordism of Witt spaces and IP spaces. Along the way, we relate the stratified and unstratified points of view for describing various classes of pseudomanifolds.

The Conner–Floyd bordism exact sequence—a new perspective

Recent work of Mishchenko and Morales Melendez (arXiv: 1112.2104 (math.AT), 2011) has shed new light on the classical exact sequence of Conner and Floyd in G-equivariant bordism for a finite group G. This paper is primarily an exposition of the results obtained there as specialized to finite groups. For a countable discrete group G, there is a brief discussion of the generalization to the bordism theory of proper, cocompact G-manifolds.

Low-dimensional surgery and the Yamabe invariant

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants \sigma(M) and \sigma(N) satisfy \sigma(N)\ge min (\sigma(M),\Lambda) for \Lambda>0. We derive explicit lower bounds for \Lambda in dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),(5,2),(6,3),(9,1),(10,1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.

ON SELF-INTERSECTION INVARIANTS

AbstractWe prove that the Hatcher–Quinn and Wall invariants of a self-transverse immersion f: Nn ↬ M2n coincide. That is, we construct an isomorphism between their target groups, which carries one onto the other. We also employ methods of normal bordism theory to investigate the Hatcher–Quinn invariant of an immersion f: Nn ↬ M2n−1.

FIXED POINTS AND COINCIDENCES IN TORUS BUNDLES

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper, we investigate fiberwise analoga and present a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace EB and a natural generalization of Nielsen numbers. As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensions over spheres and, in particular, over the unit circle. Our results are based on a careful analysis of the geometry of generic coincidence manifolds. They also allow a simple algebraic description in terms of the Reidemeister invariant (a certain selfmap of an abelian group) and its orbit behavior (e.g. the number of odd order orbits which capture certain nonorientability phenomena). We carry out several explicit sample computations, e.g. for fixed points in (S1)2-bundles. In particular, we obtain existence criteria for fixed point free fiberwise maps.

A parametrized fixed point theorem

We use bordism theory to extend Lefschetz-Nielsen theory to a family of manifolds and endomorphisms. In particular, we define an invariant, and prove a parametrized fixed point theorem and its converse.

Intersection homology with field coefficients: K‐Witt spaces and K‐Witt bordism

AbstractWe construct geometric examples of pseudomanifolds that satisfy the Witt condition for intersection homology Poincaré duality with respect to certain fields but not others. We then compute the bordism theory of K‐Witt spaces for an arbitrary field K, extending results of Siegel for K = ℚ. © 2008 Wiley Periodicals, Inc.