11月24日 (金)

13:30−14:20

蓮井 翔 (大阪府立大学), 岸本 大祐 (京都大学), 宮内 敏行 (福岡大学), 大下 顕弘 (大阪経済大学)
Samelson products in quasi-$p$-regular exceptional Lie groups
講演概要

For a compact connected simple Lie group $G$, if $G$ has no $p$-torsion in the integral homology, then there is a $p$-local homotopy equivalence $$G\simeq_{(p)}B_1\times\cdots\times B_{p-1}$$ where each $B_i$ is resolvable by spheres of dimension $2i-1\bmod2(p-1)$. If all $B_i$ are exactly spheres, then $G$ is called $p$-regular. Furthermore, if each $B_i$ is a sphere or a sphere-bundle over a sphere, then $G$ is called quasi-$p$-regular.

In studying the multiplicative structure of $G$, the Samelson products of the factor space inclusions of the above decomposition are fundamental. The (non-)triviality of these Samelson products in $p$-regular Lie groups are completely determined in our previous work, and we have recently determined the (non-)triviality in quasi-$p$-regular cases.

In this talk we first review how to determine (non-)triviality in the $p$-regular cases, and consider the quasi-$p$-regular cases after that. The (non-)triviality is completely determined by using $\mathcal{P}^1$ in the regular cases, but it is not sufficient for the quasi-regular cases. We will use a kind of secondary operations defined by using fundamental representations.

14:40−15:30

蔦谷 充伸 (九州大学), 蓮井 翔 (大阪府立大学), 岸本 大祐 (京都大学)
Higher homotopy commutativity in localized Lie groups and gauge groups
講演概要

McGibbon and Saumell studied the higher homotopy commutativity of p-localized Lie groups in the sense of Williams. In this talk, we determine the higher homotopy commutativity of p-localized Lie groups in the sense of Sugawara, which has a deep connection with higher homotopy associativity and L-S category.

This talk is based on a joint work with Sho Hasui and Daisuke Kishimoto.

15:50−16:40

若月 駿 (東京大学)
Brane topology on rational Gorenstein spaces
講演概要

We extend the loop product and the loop coproduct to the mapping space from the $k$-dimensional sphere, or more generally from any $k$-manifold, to a $k$-connected space with finite dimensional rational homotopy group, $k\geq 1$. The key to extending the loop coproduct is the fact that the embedding $M\rightarrow M^{S^{k-1}}$ is of ''finite codimension'' in a sense of Gorenstein spaces.

17:00−17:50

亀子 正喜 (芝浦工業大学)
Non-torsion non-algebraic classes in the Brown-Peterson tower
講演概要

Generalizing the classical work of Atiyah and Hirzebruch on non-algebraic classes, recently Quick proved the existence of torsion non-algebraic elements in the Brown-Peterson tower. We construct non-torsion non-algebraic elements in the Brown-Peterson tower for the prime number 2.

11月25日 (土)

09:10−10:00

入江 幸右衛門 (大阪府立大学)
Homotopy theory of polyhedral products
講演概要

$[m]=\{1,2,\cdots,m\}$ 上の単体複体 $K$ と $m$ 個の空間対の列 $(\underline{X},\underline{A})=\{(X_i,A_i)\}_{i=1}^m$ に対して polyhedral product $Z_K(\underline{X},\underline{A})$ を \[ Z_K(\underline{X},\underline{A})=\bigcup_{\sigma\in K}(\underline{X},\underline{A})^\sigma\subset X_1\times\cdots\times X_m \] と定義する。ここに、 \[ (\underline{X},\underline{A})^\sigma=Y_1\times\cdots\times Y_m \] で、$i\in \sigma$ のとき $Y_i=X_i$ で $i\not\in \sigma$ のとき $Y_i=A_i$ である。空間の列 $\underline{X}=\{X_i\}_{i=1}^m$ に対して $C\underline{X}=\{CX_i\}_{i=1}^m$ と定義する。この講演では特に、$Z_K(C\underline{X},\underline{X})$ のホモトピー型について解説を行う。研究の出発点は次の２つの結果である。

Theorem 1.1 (Bahri, Bendersky, Cohen, Gitler (2010)).

For a simplicial complex $K$ on $[m]$ and a sequence of CW-complexes $\underline{X}$ we have a homotopy equivalence \[ \Sigma Z_K(C\underline{X},\underline{X})\simeq \Sigma\left(\bigvee_{\sigma\subset[m]}\Sigma |K_I|\wedge\hat{X}^I\right), \] where $K_I=\{\sigma\subset I\ |\ \sigma\in K\}$ and $\hat{X}^I=\wedge_{i\in I}X_i $.

Theorem 1.2 (Grbić, Theriault (2007)).

If $K$ is a shifted complex on $[m]$, then we have a homotopy equivalence \[ Z_K(D^2,S^1)\simeq \bigvee_{\sigma\subset[m]}\Sigma^{|I|+1} |K_I|. \]

上記の結果から自然に次のような問題が提起された。

Problem 1.3.

定理 1.1 の結果が suspension なしに成り立つような単体複体を特徴付けろ。

この講演では、この問題に対するある程度の解決が得られたので、その結果を紹介したい。なお、これらの結果は京都大学の岸本大祐氏、および大阪府立大学の学生であった矢野達哉君との共同研究によって得られたものである。

10:20−11:10

松下 尚弘 (京都大学), 入江 幸右衛門 (大阪府立大学), 岸本 大祐 (京都大学)
Relative phantom map
講演概要

This is a joint work with Kouyemon Iriye and Daisuke Kishimoto.

A phantom map is a map between pointed CW-complexes such that its restriction to any finite dimensional subcomplex is null-homotopic. In this talk, we introduce the following generalization of phantom maps: Let X be a pointed CW-complex and $\varphi \colon B \to Y$ a map between spaces. A phantom map relative to $\varphi$ or a relative phantom map from $X$ to $\varphi$ is a map $f \to Y$ such that a restriction to any finite dimensional subcomplex of $X$ has a lift with respect to $\varphi$, up to homotopy.

We study when there is (no) non-trivial phantom maps from $X$ to $\varphi$ and give some examples. For example, there is a non-trivial relative phantom map from some space $X$ to the inclusion $j_n \colon \mathbb{R} P ^n \to \mathbb{R} P^\infty$ for $n \ge 3$, although there is no non-trivial phantom map from a suspension space to $j_n$.

11:30−12:20

小原 まり子 (信州大学), 宮内 敏行 (福岡大学), 平戸 良弘 (長野高専), 向井 純夫 (信州大学)
回転群と例外型Lie群 $G_2$ のホモトピー群について / On homotopy groups of some rotation groups and the exceptional Lie group $G_2$
講演概要

In 1967, Mimura determined the homotopy groups of Lie groups of low rank. He also showed that the 23-rd homotopy group of $Spin(7)$, $Spin(9)$, $G_2$ and $F_4$ contain the same finite group $G=Z_4$ or $(Z_2)^2$ as a direct summand. In this talk, we determine $G=Z_4$. We also determine the homotopy groups $\pi_{25}(R_n)$ by using a fiber sequence $R_{n+1} \to R_{n+1}/R_n=S^n$, where $R_n$ denotes the $n$-th rotation groups, in which we use the composition methods. We also mention the determination of $\pi_{26}(R_n)$. This is a joint work with Mukai and Hirato.

13:30−14:20

吉田 純 (東京大学)
Crossed groups and symmetries on monoidal categories
講演概要

The notion of crossed groups, originally introduced by Fiedorowicz, Loday [4], and Krasauskas [6] independently, were developed to generalize a sort of equivariances of (co)homologies. A motivational example is Connes' cyclic category $\Lambda$, which is a simplicial set defined in [2] with $\Lambda_n$ being a cyclic group of order $n+1$. It is known that the Hochschild homology admits a canonical $\Lambda$-symmetry so that we obtain the cyclic homology. On the other hand, symmetries on an algebraic structure crucially rely on those on the monoidal category it lies in, which is described in terms of operads, namely group operads, according to Corner and Gurski [3,5]. It is hence a natural question if there are any direct connection between these two notions.

In this talk, I will show that the notion of crossed interval groups, which is studied by Batanin and Markl [1], is the appropriate intermediate. More precisely, adjunctions between the category of crossed interval groups and the other two will be established. It will be seen how crossed interval groups provide notions of symmetries on monoidal categories and algebraic structures. If time permits, I will also mention the associated homology theories on algebras.

References

1. M. Batanin and M. Markl. Crossed interval groups and operations on the Hochschild cohomology. Journal of Noncommutative Geometry, 8:655-693, 2014.
2. A. Connes. Cohomologie cyclique et foncteurs $\operatorname{Ext}^n$. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 296(23):953-958, June 1983.
3. A. S. Corner and N. Gurski. Operads with general groups of equivariance, and some $2$-categorical aspects of operads in cat. arXiv:1312.5910, 2013.
4. Z. Fiedorowicz and J-L Loday. Crossed simplicial groups and their associated homology. Transactions of the American Mathematical Society, 326(1):57-87, 1991.
5. N. Gurski. Operads, tensor products, and the categorical borel construction. arXiv:1508.04050, 2015.
6. R. Krasauskas. Skew-simplicial groups. Lithuanian Mathematical Journal, 27(1):47-54, 1987.

14:40−15:30

栗林 勝彦 (信州大学)
On the category of stratifolds -- The Serre-Swan theorem --
講演概要

Stratifolds in the sense of Kreck [K] are considered from a categorical point of view. We show that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the category of smooth manifolds. Moreover, a variant of the Serre-Swan theorem for stratifolds is introduced. In particular, the category of vector bundles over a stratifold is shown to be equivalent to the category of vector bundles over an associated affine scheme although the latter is in general larger than the stratifold itself. This is joint work with Toshiki Aoki.

[K] M. Kreck, Differential Algebraic Topology, From Stratifolds to Exotic Spheres, Graduate Studies in Math., 110, AMS, 2010.

15:50−16:40

加藤 諒 (新居浜高専), 川元 祐奈 (薩南工業高校), 下村 克己 (高知大学)
On Hopkins' $E(n)$- and $K(n)$-local Picard groups
講演概要

Let $n$ be a non-negative integer $n$, and let $E(n)$ and $K(n)$ denote the $n$-th Johnson-Wilson and the $n$-th Morava $K$-theory spectra, respectively. Recently we prove that, at a prime number $p$, the Picard group of the stable homotopy category of $E(n)$-local spectra is isomorphic to a subgroup of the Picard group of the stable homotopy category of $K(n)$-local spectra. We also show that if $(p-1) \not\mid n$ and $2p-2\ge n^2$, then the $E(n)$-local Picard group consists of suspensions of $E(n)$-localized sphere spectrum, which is a generalization of a result of Hovey and Sadofsky.

17:00−17:50

南 範彦 (名古屋工業大学)
On covering by higher rational varieties -- in search for possible applications to the Morel-Voevodsky $\mathbb{A}^1$-homotopy theory

11月26日 (日)

09:40−10:30

森谷 駿二 (大阪府立大学)
The space of knots in a manifold and the right A-infinity module of configuration spaces
講演概要

In this talk, we introduce a spectral sequence converging to the singular homology of $\mathit{Emb}(S^1,M)$, the space of smooth embeddings from the circle to a closed simply connected smooth manifold $M$ of dimension $\geq 4$. Such a spectral sequence was constructed by Vassiliev. The relationship between Vassiliev's spectral sequence and ours is still unclear but if they are isomorphic at $E^2$-page, our spectral sequence could give a conprehensive description for Vassiliev's one and enable us to study it in terms of algebraic topology.

Let $STM^n$ denote $n$-times direct product of the total space of the sphere tangent bundle of $M$, and $STM^n|_X$ the total space of the restriction of the base to a subspace $X\subset M^n$. Let $F_nM\subset M^n$ denote the configuration space of $n$-ordered points in $M$, and $D_nM\subset M^n$ the fat diagonal of $M^n$. Let $\mathcal{M}_M$ denote the right $A_\infty$ module of configuration spaces of points with a tangent vector in $M$, and $\R Map(-,-)$ the derived mapping space between two right modules.

The right module $\mathcal{M}_M$ and the space $\mathit{Emb}(S^1,M)$ is related by a weak homotopy equivalence $\mathit{Emb}(S^1,M)\simeq \R Map (\mathcal{M}_{S^1}, \mathcal{M}_M)$ due to Boavida-Weiss and Turchin. For $\mathcal{M}_M$, we prove the following claim, an enriched version of the Poincar\'e-Lefschetz duality $H_*(STM^n|_{F_nM})\simeq H^*(STM^n, STM^n|_{D_nM})$:

Theorem 1. Let $M$ be a closed orientable smooth manifold. In the category of symmetric spectra, the dual comodule of $\mathcal{M}_M$ is weakly equivalent to a comodule $C_M$ consisting of certain Thom spectra associated to a vector bundle over $STM^n/(STM^n|_{D_nM})$.

The comodule $C_M$ in Theorem 1 has a $\check{\text{C}}$ech-type resolution, and it induces the desired spectral sequence:

Theorem 2. Let $R$ be a commutative ring and $M$ a closed simply connected smooth manifold of dimension $\geq 4$. There exists a spectral sequence converging to $H_*(\mathit{Emb}(S^1,M);R)$ whose $E^2$-page is described by the homology groups of a fiber product $STM^n\times_{M^n}M^k$ for various $n,k$ with $n>k$ and the diagonal maps and the shriek maps between them.

Though our main concern is the ordinary homology, the use of symmetric spectra is technically inevitable. Our construction is similar to that of the Cohen-Jones isomorphism in string topology, and we need to deal with higher homotopy concerning shriek maps. Symmetric spectra is suitable to this kind of problem.

10:50−11:40

岸本 大祐 (京都大学), S. Theriault (University of Southampton)
Mod $p$ homology of the classifying space of a gauge group
講演概要

Let $G$ be a compact connected simple Lie group of type $n_1 < \cdots < n_\ell$. We calculate the mod $p$ homology of the classifying space of a gauge group of a principal $G$-bundle over $S^4$ with the second Chern class having the value $k\in\mathbb{Z}$ which is not divisible by $p$. The main tool is Tsutaya's delooping of the evaluation fibration. This is joint work with Stephen Theriault.