Algebraic Topology

  title={Algebraic Topology},
  author={Raphael Ho}
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology. 

Homology and homotopy groups of the complement of certain family of fibers in a critical point problem.

In this paper we improve the results of [5] by showing that the collection of fibers over a closed countable subset of the base space of a dierentiable fibration

Derived Algebraic Geometry III: Commutative Algebra

This paper describes a higher-categorical version of the theory of colored operads, giving applications to the study of commutative ring spectra.

Smooth covers of finite groups.

In the spirit of the homology theory where algebraic and geometric concepts merge, we establish that a natural order preserving condition for covering groups corresponds to having a smooth covering

Commutativeness of Fundamental Groups of Topological Groups

It is proved that fundamental groups based at the unit point of topological groups are commutative and the inequality of the following types of groups is solved.

A Spectral Perspective on Neumann-Zagier

We provide a new topological interpretation of the symplectic properties of glu- ing equations for triangulations of hyperbolic 3-manifolds, rst discovered by Neumann and

The Comparison of K-Theory and TC

This chapter begins with a proof that the difference between algebraic K-theory and topological cyclic homology is “locally constant” by means of a hands-on approach, nesting our way down from the

On the Cohomology of the Complement of a Toral Arrangement

The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.

Automorphisms of Free Groups and Outer Space

This is a survey of recent results in the theory of automorphism groups of finitely-generated free groups, concentrating on results obtained by studying actions of these groups on Outer space and its

Treatment of the cyclic homology through topological spaces

- The purpose of this paper is proving the analysis of vanishing and non-vanishing cyclic and reflexive (co)homology groups of algebras. Additionally, we present and prove some of the results and