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| 研究生： |
林育萱 Yu-Hsuan Lin |
|---|---|
| 論文名稱： | Lefschetz Fixed Point Formula on Absolute and Relative de Rham Complexes |
| 指導教授： |
黃榮宗
Rung-Tzung Huang |
| 口試委員: | |
| 學位類別： |
碩士 Master |
| 系所名稱： |
理學院 - 數學系 Department of Mathematics |
| 論文出版年： | 2024 |
| 畢業學年度： | 112 |
| 語文別： | 英文 |
| 論文頁數： | 27 |
| 中文關鍵詞： | Lefschetz 固定點公式 、de Rham 複形 、邊界條件 、熱核方法 、固定點 、Lefschetz 數 |
| 外文關鍵詞： | Lefschetz fixed point formula, de Rham complex, Boundary condition, Heat kernel method, Fixed point, Lefschetz number |
| 相關次數： | 點閱：16 下載：0 |
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此論文運用熱核方法，研究僅具有簡單固定點、且保持邊界的光滑映射在絕對和相對 de Rham 複形上的Lefschetz 固定點公式。通過同倫變形和構造參數熱核來推導公式，並同時考慮內部及邊界的固定點。我們的結果為 Lefschetz 固定點公式提供了熱核證明，闡釋邊界條件在 de Rham 複形上的作用。
In this thesis, we use the method of heat kernels to study the Lefschetz fixed point formula on absolute and relative de Rham complexes with respect to a smooth boundary-preserving map with simple fixed points only. We employ homotopy deformation and construct parametrix heat kernels to derive the formula, accounting for both interior and boundary fixed points. Our results provide a heat kernel proof of the Lefschetz fixed point formula, elucidating the role of boundary conditions in de Rham complexes.
[1] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. Math. 86 (1967),
374-407.
[2] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc.
Cambridge Philos. Soc. 77 (1975), 43-69.
[3] F. Bei, The L2-Atiyah-Bott-Lefschetz theorem on manifolds with conical singularities: a heat kernel approach,
Ann. Glob. Anal. Geom. 44 (2013), 565-605.
[4] V. A. Brenner and M. A. Shubin, The Atiyah-Bott-Lefschetz formula formula for elliptic complexes on
manifolds with boundary, J. Soviet Math. 64 (1993), 1069-1111.
[5] B. Booβ-Bavnbek and K. Wojciechowski, Elliptic Boundary Value Problems for Dirac Operators, Birkhäuser,
Boston, 1993.
[6] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd Edition, CRC
Press, Inc., 1994.
[7] R.-T. Huang and Y. Lee, The refined analytic torsion and a well-posed boundary condition for the odd signature
operator, J. of Geom. and Phy., Vol 126, (2018), 68-92.
[8] R.-T. Huang and Y. Lee, Lefschetz fixed point formula on a compact Riemannian manifold with boundary for
some boundary conditions, Geometriae Dedicata, 181, Issue 1, (2016), 43-59.
[9] S. Klimek and K. Wojciechowski, Adiabatic Cobordism Theorems for Analytic Torsion and η-Invariant, J.
Funct. Anal., 136, 269-293 (1996).
[10] J. Roe, Elliptic operators, topology and asymptotic methods, 2nd Edition, Research Notes in Mathematics series
395, Chapman and Hall/CRC, 1998.
