COMPUTATIONAL METHODS IN ALGEBRAIC TOPOLOGY

代数拓扑中的计算方法

吴一凡

11930006

硕士

数学

朱一飞

2021-05-20

2021-05-20

南方科技大学

深圳

This thesis is motivated by a classification problem of continuous maps between topological spaces up to homotopy. For this purpose, it then turns to some computational methods that are useful albeit complicated, namely, spectral sequences and cohomology operations, and uses them to calculate with some interesting examples. Chapter 3 introduces the general construction and properties of a spectral sequence, and specializes these ideas to the Serre spectral sequences for homology and cohomology with explicit constructions. We calculate $\pi_4(S^3)$ and reprove the finiteness of stable homotopy groups in positive dimensions through an argument with the Serre classes. Chapter 4 introduces a class of important operations on cohomology, carrying a lot of extra information and structures than classical cohomology theories. They form the Steenrod algebras $\mathcal{A}_p$ for $p$ a prime number. Their applications include giving an upper bound for orthogonal tangent vector fields on spheres and Adams spectral sequences. The latter becomes an extremely powerful tool in algebraic topology, calculating the stable homotopy classes of maps between two topological spaces. As an example, we calculate the 2-component of $\pi_2^s$. The last two sections of Chapter 4 introduce K-theory as a generalized cohomology theory and the Adams operations as its cohomology operations. Using them we deduce two classical facts in algebraic topology, namely, Adams's theorem on Hopf invariants and the classification of finite-dimensional division algebras over $\mathbb{R}$.

这篇硕士论文启发自拓扑空间之间的连续映射在同伦意义下的分类问题，由此引出针对这种问题的计算方法的探究。着重介绍了两种重要尽管繁琐的计算工具，分别是谱序列和上同调运算，并运用这些工具对一些例子进行了计算。论文的第三章主要介绍了谱序列的构造和基本性质，研究了一种特别的谱序列——塞尔同调及上同调谱序列。运用塞尔上同调谱序列重新计算了三维球面的四维同伦群，之后通过引入塞尔类重新证明了除0维外所有球面稳定同伦群都是有限交换群。第四章的前半部分介绍了上同调运算，特别是斯廷罗德构造的斯廷罗德代数。它将同调函子从拓扑空间范畴到环范畴推广为到斯廷罗德代数上的模范畴，而且保留了之前的环结构。论文运用斯廷罗德代数重新计算了球面上正交切向量场的最大个数的一个上界。另一个斯廷罗德代数的应用是亚当斯谱序列，用于计算两个拓扑空间之间的由映射的稳定同伦类构成的群，以球面二维稳定同伦群的2分量的实际计算作为应用。第四章的后半部分独立于前半部分介绍了一种广义上同调理论——K理论，以及它上面附带的上同调运算。运用它们可以得到代数拓扑中关于霍普夫不变量为1的经典定理以及实线性空间作为实数域上的可除代数的结论。

Serre spectral sequence Adams spectral sequence cohomology operation stable homotopy group K-theory

塞尔谱序列 亚当斯谱序列 上同调运算 稳定同伦群 K理论

英语

独立培养

南方科技大学

Wu YF. COMPUTATIONAL METHODS IN ALGEBRAIC TOPOLOGY[D]. 深圳. 南方科技大学,2021.