Michael A. Hill
We compute the low dimensional String bordism groups of $BE_8$ and $BE_8\times BE_8$ using a combination of Adams spectral sequences together with comparisons to the Spin bordism cases.
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Bordism for the 2-group symmetries of the heterotic and CHL strings
本文研究了非零B场下heterotic string和CHL string的2-group对称性，将其bordism群表示为扭曲string bordism群并计算至11维，进而用于研究这两种string theory的反常与缺陷。
This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle characteristic numbers arising in Borel-Hirzebruch Lie-group-theoretic calculations correspond precisely to the divisibility properties arising in the Hovey-Ravenel-Wilson BP-Hopf-ring-theoretic calculation of string bordism at primes >3.
We show that $K(2)$-locally, the smash product of the string bordism spectrum and the spectrum $T_2$ splits into copies of Morava $E$-theories. Here, $T_2$ is related to the Thom spectrum of the canonical bundle over $ΩSU(4)$.
We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes $ζ_n$ take the places of the prime numbers, and this allows us to discuss higher bordism theories. We prove that the K(2)-localizations of the spectrum of topological modular forms as well as the string bordism spectrum have characteristic $ζ_2$.
Using spectral invariants of Dirac operators we construct a secondary version of the Witten genus, a bordism invariant of string manifolds in dimensions $4m-1$. We prove a secondary index theorem which relates this global-analytic construction with its homotopy-theoretic analog. The latter will be calculated through its factorization over topological modular forms.
Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extending previous results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the prop of these reduced diagrams. Some of these operations are extended to spaces of maps from a sphere to a compact manifold.
Global anomalies & bordism of non-supersymmetric strings
针对十维非超对称弦理论的全局反常一致性问题，通过Adams谱序列计算扭曲弦配边群，证明两类模型配边群完全消失、一类部分消失，还给出低维配边群结果与NS5-膜世界体构造，副产品为重新验证杂化弦外自同构无反常。
In this work we investigate the Swampland Cobordism Conjecture in the context of type IIB string theory geometries with non-trivial duality bundle. Quite remarkably, we find that many non-trivial bordism classes with duality bundles in Mp$(2,\mathbb{Z})$, a double cover of SL$(2,\mathbb{Z})$ related to fermions, correspond to asymptotic boundaries of well-known supersymmetric F-theory backgrounds. These include $[p,q]$-7-branes, non-Higgsable clusters, S-folds, as well as various lower-dimensional generalizations. These string theoretic objects break the global symmetries associated to the non…
This paper shows that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}\langle8\rangle[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of $π_*\mathrm{MO}\langle8\rangle$, the $π_*\mathrm{MO}\langle8\rangle$-module $$π_*\big(\mathrm{MO}\langle8\rangle_{(p)}/X\big)$$ is not (even abstractly) isomorphic to $π_*\mathrm{tmf}_{(p)}$. It does so by showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of $π_*(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces $$π_*(R/X)\otimes\mathbf{Q} \cong …
Stable Cohomotopy in Codimensions Two and Three: From Algebraic Characterizations to Bordism-Theoretic Interpretations
本文从代数和几何角度研究二维和三维余维数中的稳定余余维数，给出了二维的完整特征和三维的3-原参数特征，并提供了定向流形和弦流形的 bordism 解释，推导了向量丛无处为零截面存在的充要条件。
Computations of Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7
本文利用谱序列、障碍理论和上同调运算，计算了维度至多 7 时 Spin-$G$（$G$ 为 $Sp(4)$、$SU(8)$、$Spin(16)$）配边群的显式生成元，揭示了 Spin 与额外对称性的相互作用。
For each space X we define an explicit group, G(X), functorially in X. This group is constructed from the groups of cochains on X. Furthermore, we construct an explicit functorial pairing with values in R/Z between the cochain representatives for elements of G(X) and maps of closed 3-dimensional spin manifolds to X. This pairing induces a pairing between G(X) and the 3-dimensional spin bordism group of X and identifies each with the Pontrjagin dual of the other.
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-, and 3-manifolds is the free symmetric monoidal bicategory on a single anomaly-free modular object.
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.
The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring on even-dimensional generators. In this paper we review the cases in which the conjecture is known to hold and we highlight the properties that permit one to prove the conjecture in these cases.
We consider a category whose morphisms are bordisms of $n$-dimensional pseudomanifolds equipped with a certain additional structure (coloring). On the other hand, we consider the product $G$ of $(n+1)$ copies of infinite symmetric group. We show that unitary representations of $G$ produce functors from the category of $(n-1)$-dimensional bordisms to the category of Hilbert spaces and bounded linear operators.
In the present paper we construct two new explicit complex bordisms between any two projective bundles over $\mathbb{C} P^1$ of the same complex dimension, including the Milnor hypersurface $H_{1,n}$ and $\mathbb{C} P^1\times \mathbb{C} P^{n-1}$. These constructions reduce the bordism problem to the null-bordism of some projective bundle over $\mathbb{C} P^1$ with the non-standard stably complex structure.
We classify invertible 2-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. The zeroth homotopy group of a bordism category is the usual Thom bordism group, the first homotopy group can be identified with a Reinhart vector field bordism group, or the so called SKK group as observed by Ebert, Böckstedt-Svane and Kreck-Stolz-Teichner. We present the computation of SKK groups for stable tangential structures. Then we consider non-stable examples: the 2-dimensional framed and $r$-spin SKK groups…
Given a monoidal $\infty$-category $C$ equipped with a monoidal recollement, we give a simple criterion for an object in $C$ to be dualizable in terms of the dualizability of each of its factors and a projection formula relating them. Predicated on this, we then characterize dualizability in any monoidally stratified $\infty$-category in terms of stratumwise dualizability and a projection formula for the links. Using our criterion, we prove a 1-dimensional bordism hypothesis for symmetric monoidal recollements. Namely, we provide an algebraic enhancement of the 1-dimensional framed bordism $\i…
Equivariant bordism classification of five-dimensional $(\mathbb{Z}_2)^3$-manifolds with isolated fixed points
本文针对具有有效光滑$(\mathbb{Z}_2)^3$-作用且不动点孤立的五维闭光滑流形，确定其等变无定向配边群$\mathcal{Z}_5((\mathbb{Z}_2)^3)$的$\mathbb{Z}_2$维数为77并给出基，完成该类流形的等变无定向配边分类。
The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $Ω^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms wit…
This note provides a computation of the bordism groups of K-Witt spaces for fields K with characteristic 2. We provide a complete computation for the unoriented bordism groups. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambiguity in dimensions equivalent to 2 mod 4. This corrects an error in the char(K)=2 case of the author's prior computation of the bordism groups of K-Witt spaces for an arbitrary field K.
We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected manifolds with positive 2-curvature in terms of the spin and string bordism groups, and by means of $α$-invariant and Wit…
Cobordism Utopia: U-Dualities, Bordisms, and the Swampland
本研究围绕沼泽地配边猜想，计算不同维度超引力理论的U-对偶配边群，确定对应弦论、M/F-理论的对偶缺陷背景，还发展了针对SL(2,Z)×SL(3,Z)配边群的新计算方法。
Associated with the Davis-Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the "dual" algebra of the unoriented $G_n$-representation algebra introduced by Conner and Floyd, where $G_n=(\Z_2)^n$. With the help of $G_n$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G_n$-equivariant unoriented …
Quantitative asphericalization and Gromov's linear bordism problem
本文证明Gromov线性猜想的PL配边版本对广泛球面化群成立，并利用定量球面化方法将PL (4k-1)-流形的$L^2$ $\rho$-不变量界从超阶乘增长改进为阶乘增长。
We give bordism-finiteness results for manifolds with semi-simple group action. Consider the class of oriented manifolds which admit a circle action with isolated fixed points such that the action extends to an $S^3$-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontrjagin class, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with $S^3$-action as above.
Equivariant Milnor map
将Milnor映射推广到等变情形，证明圆群和二阶循环群的等变酉群上存在该映射，并关联其核与自由共轭的磁等变酉配边群。
Oriented and unitary equivariant bordism of surfaces
本研究显式计算了有限群$G$作用下光滑紧曲面的定向与酉等变配边群，刻画了其秩与挠子群的结构，并给出相关配边性质的新证明，建立了二维SK群与等变配边群的联系。
We construct spaces of 1-dimensional supersymmetric Euclidean field theories and show that they represent real or complex K-theory. A noteworthy feature of our bordism category is that the identity bordism of a point is connected to intervals of positive length.
Understanding Non-Split 2-Group Symmetry: (3+1)D SymTFT, Anomaly and Bordism
该研究系统分析含ℤ₂ 0-形式与1-形式对称性的非分裂2-群𝒢，计算d≤5时的配边群以分类其反常，在d=3时结合融合2-范畴与TQFT作用研究其Symmetry TFT并分类边界条件，实施范畴化朗道范式。
Let $M$ be an oriented manifold and let $\frak N$ be a set consisting of oriented closed manifolds of the same odd dimension. We consider the topological space $G_{\frak N, M}$ of commutative diagrams. Each commutative diagram consists of a few manifolds from $\frak N$ that are mapped to $M$ and a few one point spaces $pt$ that are each mapped to a pair of manifolds from $\frak N$. We consider the oriented bordism group $Ω_*(G_{\frak N, M})=\oplus_{i=0}^{\infty} Ω_i(G_{\frak N, M}).$ We introduce the operations $\star$ and $[\cdot, \cdot]$ on $Ω_*(G_{\frak N,M})\otimes \mathbb Q,$ that make $Ω…
Quantitative bordism over acyclic groups and Cheeger-Gromov $ρ$-invariants
本文研究了acyclic群的量化bordism问题，并证明了PL(4k-1)流形的Cheeger-Gromov $ρ$-不变量存在普遍线性界，同时揭示了具有固定简单同调类型的流形的三角剖分复杂性。
Characteristic numbers of canonical toric manifolds and their applications
本文计算了典范胞腔流形的陈数、米尔诺数和庞特里亚金数，及其实数对应物的斯蒂夫尔-惠特尼数，探讨了其在单位ary、定向和非定向 bordism 类的组合刻画、单位ary bordism 环生成元的几何代表、典范胞腔流形的边界判定准则以及其浸入欧几里得空间的维数估计。
We prove a theorem in 3-dimensional topological field theory: a Reshetikhin-Turaev theory admits a nonzero boundary theory iff it is a Turaev-Viro theory. The proof immediately implies a characterization of fusion categories in terms of dualizability. The main theorem applies to physics, where it implies an obstruction to a gapped 3-dimensional quantum system admitting a gapped boundary theory. Appendices on bordism multicategories and on internal duals may be of independent interest.; v2 extensive revision: added theorem on dualizable 2-categories, material on natural transformations, reworke…
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which 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-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.
For a finite group $G$, we define the $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an isomorphism of its fundamental group onto the direct sum $\mathbb{Z}\oplus Ω_2^{SO}(BG)$, where $Ω_2^{SO}(BG)$ is the free oriented $G$-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product $G/[G,G]\times S^1\times X^G$, where $π_1(X^G)=Ω_2^{SO}(BG)$. Fin…
We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems…
We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by na…
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