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\leftline{Yuka Umemoto}
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\noindent
Part ${\rm I}$:  Algebraic Topology

\noindent
References :

\noindent
1. Bredon, Glen : \underbar {Topology and Geometry}

\noindent
2. Greenberg, Marvin and John Harper : \underbar {Algebraic Topology,
a First Course}

\noindent
3. Rolfsen, Dale : \underbar {Knots and Links}



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\noindent
The following are taken from the references.

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\noindent
\underbar {Fundamental Group} (Bredon, Chapter $\rm III$)

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1. Homotopy Groups

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2. The Fundamental Group

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3. Covering Spaces

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4. The Lifting Theorem

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5. The Action of $\pi_1$ on the Fiber

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6. Deck Transformations

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7. Properly Discontinuous Actions

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8. Classification of Covering Spaces

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9. The Seifert-Van Kampen Theorem

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\noindent
\underbar {Singular Homology Theory} ( Greenberg and Harper )

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8. Affine Preliminaries

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9. Singular Theory

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10. Chain Complexes

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11. Homotopy Invariance of Homology

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12. Relation Between $\pi_1$ and $H_1$

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13. Relative Homology

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14. The Exact Homology Sequence

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15. The Excision Theorem

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16. Further Applications to Spheres

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17. Mayer-Vietoris Sequence

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18. The Jordan-Brower Separation Theorem

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19. Construction of Spaces: Spherical Complexes

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20. Betti Numbers and Euler Characteristic

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21. Constuction of Spaces: Cell Complexes and More Adjunction Spaces

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\noindent
\underbar {Orientation and Duality on Manifolds} ( Greenberg and Harper )

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22. Orientation of Manifolds

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23. Singular Cohomology

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24. Cup and Cap Products

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25. Algebraic Limits

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26. Poincare Duality

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27. Alexander Duality

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28. Lefschetz Duality 

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\noindent
\underbar {Fundamental Group and Alexander Invariant of Knots} (Rolfsen) 

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Chapter 3: The Fundamental Group 

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A. Knot and link invariants

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B. The knot group

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C. Torus knots

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D. The Wirtinger presentation

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Chapter 5: Seifert Surfaces

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A. Surfaces and genus

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C. Construction of the cyclic coverings of a knot complement using Seifert
surfaces

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\noindent
Chapter 6: Finite Cyclic Coverings and Torsion Invariants

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B. Calculation using Seifert surfaces

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Chapter 7: Finite Cyclic Coverings and the Alexander Invariant

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A. The Alexander invariant

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B. Seifert surfaces again

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Chapter 10: Foliations, Branched covers, Fibrations and so on

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H. Fibered knots and links

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I. Fibering the complement of a trefoil

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Part ${\rm II}$: Differential Geometry
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\noindent
Reference: 

\noindent
Boothby, William : \underbar {An Introduction to
Differentiable Manifolds and Riemannian Geometry}
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\noindent 
The following are taken from the reference.  
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\noindent
Chapter ${\rm III}$: Differentiable Manifolds and Submanifolds

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1. The Definition of a Differentiable Manifold

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2. Further Examples

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3. Differentiable Functions and Mappings

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4. Rank of a Mapping. Immersions

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5. submanifolds

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6. Lie Groups

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7. The Action of a Lie Group on a Manifold.  Transformation Groups

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8. The Action of a Discrete Group on a Manifold

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9. Covering Manifolds 

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\noindent
Chapter ${\rm IV}$: Vector Fields on a Manifold

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1. The Tangent Space at a Point of a Manifold

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2. Vector Fields

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3. One-Parameter and Local One-Parameter Groups Acting on a Manifold

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4. The Existence Theorem for Ordinary Differential Equations

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5. Some Examples of One-Parameter Groups Acting on a Manifold

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6. One-Parameter Subgroups of Lie Groups

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7. The Lie Algebra of Vector Fields on a Manifold

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8. Frobenius' Theorem

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9. Homogeneous Spaces


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\noindent
Chapter ${\rm V}$: Tensors and Tensor Fields on Manifolds

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1. Tangent Covectors

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2. Bilinear Forms. The Riemannian Metric

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3. Riemannian Manifolds as Metric Spaces

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4. Partition of Unity

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5. Tensor Fields

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6. Multiplication of Tensors

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7. Orientation of Manifolds and the Volume Element

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8. Exterior Differentiation

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\noindent
Chapter ${\rm VII}$: Differentiation on Riemannian Manifolds

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1. Differentiation of Vector Fields along Curves in $\bf R^n$

The Geometry of Space Curves

Curvature of Plane Curves

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2. Differentiation of Vector Fields on Submanifolds of $\bf R^n$

Formulas for Covariant Derivatives

$\nabla _X Y$ and Differentiation of Vector Fields

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3. Differentiation on Riemannian Manifolds

Constant Vector Fields and Parallel Displacement

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4. The Curvature Tensor
The Riemannian Connection and Exterior Differential Forms

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5. Geodesic Curves on Riemannian Manifolds

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6. The Tangent Bundle and Exponential Mapping.  Normal Coordinates

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\noindent 
Chapter ${\rm VIII}$: Curvature

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1. The Geometry of Surfaces in $\bf E^3$

Principle Curvatures at a Point of a Surface

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2. The Gaussian and Mean Curvatures of a Surface

The Theorema Egregium of Gauss

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3. Basic Properties of the Riemannian Curvature Tensor

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4. The Curvature Forms and the Equations of Structure

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5. Differentiation of Covariant Tensor Fields

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6. Manifolds of Constant Curvature

Spaces of Positive Curvature

Spaces of Zero Curvature

Spaces of Constant Negative Curvature

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