Problem 3: Homotopy and Fundamental Group Calculations Let X be a torus T², and consider the following loops on T²: • Loop a runs around the "longitude" of the torus. • Loop ẞ runs around the "latitude" of the torus. 1. Compute the fundamental group π1 (7², xo), where x is a base point, using generators and relations. 2. Show that any loop on 72 can be expressed as a combination of a and ẞ, and prove that the group is isomorphic to Z × Z. → 3. Consider a continuous map f: T² S¹ (where S¹ is the unit circle) that "wraps" the torus around the circle. Describe the induced map on the fundamental group f* : π₁(T²) → T1 (S1) and discuss its kernel and image. 4. Provide a graphical representation of the loops a, ẞ, and an example of a homotopy on the torus. Problem 4: Spectral Theory in Functional Analysis Let T : X → X be a bounded linear operator on a Banach space X, with the spectrum of T denoted by σ (T). 1. Prove that if T is compact, then every non-zero element of the spectrum σ (T) is an eigenvalue of T.

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Chapter4: More On Groups
Section4.5: Normal Subgroups
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Problem 3: Homotopy and Fundamental Group Calculations
Let X be a torus T², and consider the following loops on T²:
•
Loop a runs around the "longitude" of the torus.
•
Loop ẞ runs around the "latitude" of the torus.
1. Compute the fundamental group π1 (7², xo), where x is a base point, using generators and
relations.
2. Show that any loop on 72 can be expressed as a combination of a and ẞ, and prove that the
group is isomorphic to Z × Z.
→
3. Consider a continuous map f: T² S¹ (where S¹ is the unit circle) that "wraps" the torus
around the circle. Describe the induced map on the fundamental group f* : π₁(T²) →
T1 (S1) and discuss its kernel and image.
4. Provide a graphical representation of the loops a, ẞ, and an example of a homotopy on the
torus.
Problem 4: Spectral Theory in Functional Analysis
Let T : X → X be a bounded linear operator on a Banach space X, with the spectrum of T
denoted by σ (T).
1. Prove that if T is compact, then every non-zero element of the spectrum σ (T) is an
eigenvalue of T.
Transcribed Image Text:Problem 3: Homotopy and Fundamental Group Calculations Let X be a torus T², and consider the following loops on T²: • Loop a runs around the "longitude" of the torus. • Loop ẞ runs around the "latitude" of the torus. 1. Compute the fundamental group π1 (7², xo), where x is a base point, using generators and relations. 2. Show that any loop on 72 can be expressed as a combination of a and ẞ, and prove that the group is isomorphic to Z × Z. → 3. Consider a continuous map f: T² S¹ (where S¹ is the unit circle) that "wraps" the torus around the circle. Describe the induced map on the fundamental group f* : π₁(T²) → T1 (S1) and discuss its kernel and image. 4. Provide a graphical representation of the loops a, ẞ, and an example of a homotopy on the torus. Problem 4: Spectral Theory in Functional Analysis Let T : X → X be a bounded linear operator on a Banach space X, with the spectrum of T denoted by σ (T). 1. Prove that if T is compact, then every non-zero element of the spectrum σ (T) is an eigenvalue of T.
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