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Mitsunobu Tsutaya, Homotopy type of the space of finite propagation unitary operators on Z, Southampton-Kyoto Workshop II, 2020.12, The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that π0 of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this talk, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable C-valued Z-sequences, so we can determine its homotopy groups.. |

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Mitsunobu Tsutaya, Homotopy types of spaces of finite propagation unitary operators on Z, WORKSHOP: unitary operators: spectral and topological properties, 2020.09. |

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Unstable homotopy types of spaces of finite propagation unitary operators on Z. |

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Mitsunobu Tsutaya, Characterizations of homotopy fiber inclusion, Homotopy Theory Symposium, 2019.11, It is well-known that a homotopy fiber sequence generated by a map G --> X extends twice to the right if and only if G admits a structure of a topological group and the map extends to an action of G on X up to homotopy equivalence. Similarly, a homotopy fiber sequence generated by a map H --> G extends three times to the right if and only if H and G admit structures of topological groups and the map is a homomorphism up to homotopy equivalence. But characterizations of "homotopy fiber inclusions" do not seem to be studied in detail. In this talk, we show some characterizations of homotopy fiber inclusions and some examples.. |

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Mitsunobu Tsutaya, A_{n}-maps and mapping spaces, Mapping Spaces in Algebraic Topology, 2018.08, An-maps are morphisms between An-spaces introduced by Sugawara, Stasheff, Boardman-Vogt and Iwase. Sugawara, Stasheff and Iwase characterised the condition when a map between An-spaces admits an An-map structure in terms of projective spaces. In this talk, we see that a refinement of this result is realised as a weak homotopy equivalence between the space of An-maps An(G,H) and the space of based maps Map∗(BnG,BH) from the n-th projective space BnG to the classifying space BH. We also see some applications of this results to extension of an evaluation fibration and homotopy commutativity.. |

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Mitsunobu Tsutaya, Mapping spaces from projective spaces, International Conference on Manifolds, Groups and Homotopy, 2018.06. |

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Mitsunobu Tsutaya, Homotopy theoretic classifications of gauge groups, Young Researchers in Homotopy Theory and Toric Topology 2017, 2017.08. |

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Mitsunobu Tsutaya, Infiniteness of A∞-types of gauge groups, Friday's Topology Seminar, 2017.02. |

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Mitsunobu Tsutaya, Higher homotopy commutativity in localized Lie groups and gauge groups, Topology & Malaga Meeting, 2017.02. |

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The A_n-structure (n=1,2,...,∞) of a topological group describes certain higher homotopy structure concerned with its binary operation. It has some relation with a generalization of projective spaces and homotopy invariants such as LS-category. The notion. |