Geometric Topology: Master the Interplay of Geometry and Topology

Question 1

Which of the following is a consequence deduced from the Gauss-Bonnet theorem?

Accepted Answer

The total curvature of a closed surface is a topological invariant.

Answer

The Euler characteristic of a closed surface is determined by its curvature.

Answer

Any two closed surfaces with equal curvature are diffeomorphic.

Answer

A surface's Gaussian curvature is invariably positive.

Question 2

Which of the following is NOT a type of geometric 3-manifold?

Accepted Answer

Symplectic manifold

Answer

Haken manifold

Answer

Seifert manifold

Answer

Hyperbolic manifold

Question 3

What is the fundamental group of a Klein bottle?

Accepted Answer

ℤ/2ℤ

Answer

ℤ

Answer

ℤ⊕ℤ

Answer

Free group on 2 generators

Question 4

Poincaré duality states that for a compact n-manifold:

Accepted Answer

H_i(M) is isomorphic to H^{n-i}(M)

Answer

H^i(M) is isomorphic to H_{n-i}(M)

Question 5

In the study of low-dimensional manifolds, what characterizes a 2-manifold with a boundary?

Accepted Answer

A surface with an edge

Answer

A surface without an edge

Answer

A closed surface

Answer

A three-dimensional space

Question 6

For a circle, which group describes its fundamental group, the group of all closed paths based at a point?

Accepted Answer

Infinite cyclic group

Answer

Trivial group

Answer

Free group on two generators

Answer

Symmetric group on three elements

Question 7

In geometric topology, which homology theory assigns algebraic structures to geometric objects?

Accepted Answer

Singular homology

Answer

Betti homology

Answer

Group homology

Answer

Čech homology

Question 8

Which surface results from gluing two Möbius strips, one-sided surfaces, together?

Accepted Answer

Klein bottle

Answer

Sphere

Answer

Torus

Answer

Projective plane

Question 9

The Poincaré duality theorem for a 3-manifold, a three-dimensional space, establishes a duality between which homology groups?

Accepted Answer

H_n(M) and H^(3-n)(M)

Answer

H_n(M) and H^(3+n)(M)

Answer

H_n(M) and H_(3-n)(M)

Answer

H_n(M) and H_(3+n)(M)

Question 10

The Thurston Geometrization Conjecture aims to classify compact 3-manifolds based on which geometric structures?

Accepted Answer

Euclidean, spherical, hyperbolic, and Seifert-Weber

Answer

Poincaré, Dehn, and SnapPea

Answer

Hodge, Kähler, and Calabi-Yau

Answer

abc, XYZ, and 123

Question 11

What geometrically represents the fundamental group of a surface?

Accepted Answer

All closed paths on the surface, equivalent up to homotopy.

Answer

All non-contractible closed paths on the surface.

Answer

All loops on the surface.

Question 12

Which of the following is **not** a topological property of a knot?

Accepted Answer

Colorability

Answer

Knot type

Answer

Triviality

Answer

Unknotting number

Question 13

Which of the following is a characteristic property of hyperbolic manifolds?

Accepted Answer

Negative curvature

Answer

Positive curvature

Answer

Zero curvature

Answer

Flat geometry

Question 14

Which property of a knot remains unchanged under any type of continuous deformation?

Accepted Answer

Alexander polynomial

Answer

Genus

Answer

Euler characteristic

Answer

Betti number

Question 15

What is the first homology group of a circle?

Accepted Answer

Z

Answer

Z^2

Answer

Z_2

Answer

Z^3

Question 16

Which group describes the path-connected components of the 2-torus?

Accepted Answer

Z^2

Answer

Z

Answer

Z_2

Answer

Z_3

Question 17

What is the mathematical operation in Dehn surgery that involves removing a tubular neighbourhood around a knot and then reattaching it with a twist?

Accepted Answer

Removing a tubular neighbourhood and twisting

Answer

Adding a handle

Answer

Slicing the knot into two pieces

Answer

Projecting the knot onto the plane

Question 18

Which property describes a knot invariant?

Accepted Answer

Knot type

Answer

Knot crossings

Answer

Knot length

Answer

Knot projection

Question 19

Which of the following is a hyperbolic 3-manifold?

Accepted Answer

A closed orientable 3-manifold with an infinite fundamental group

Answer

The 3-sphere

Answer

A lens space

Answer

A closed orientable 3-manifold with a finite fundamental group

Question 20

What is a PL manifold?

Accepted Answer

A manifold that can be locally approximated by polyhedra

Answer

A manifold that is homeomorphic to a smooth manifold

Answer

A manifold that is orientable

Answer

A manifold that is compact