Problem 32 - Algebraic Topology

Question

Consider covering spaces \(\boldsymbol{p}: \tilde{\boldsymbol{X}} \rightarrow X\) with \(\tilde{\boldsymbol{X}}\) and \(X\) connected CW complexes, the cells of \(\tilde{X}\) projecting homeomorphically onto cells of \(X\). Restricting \(p\) to the 1-skeleton then gives a covering space \(\tilde{X}^{1} \rightarrow X^{1}\) over the 1 -skeleton of \(X .\) Show: (a) Two such covering spaces \(\tilde{X}_{1} \rightarrow X\) and \(\tilde{X}_{2} \rightarrow X\) are isomorphic iff the restrictions \(\tilde{x}_{1}^{1} \rightarrow x^{1}\) and \(\tilde{X}_{2}^{1} \rightarrow X^{1}\) are isomorphic. (b) \(\widetilde{x} \rightarrow X\) is a normal covering space iff \(\tilde{X}^{1} \rightarrow X^{1}\) is normal. (c) The groups of deck transformations of the coverings \(\tilde{X} \rightarrow X\) and \(\tilde{X}^{1} \rightarrow X^{1}\) are isomorphic, via the restriction map.

Step-by-Step Solution

Answer

(a) Isomorphic if 1-skeleton restrictions are isomorphic. (b) Normal if 1-skeleton is normal. (c) Deck transformation groups are isomorphic.

Step 1: Understanding covering space restriction

Begin by considering the definition of covering spaces and the restriction to 1-skeletons, specifically the covering space transformations needed to show isomorphisms.

Step 2: Establish Homeomorphism on Cells

Note that the covering map between higher dimensions is dictated by its effect on the 1-skeleton due to the homeomorphic mapping of cells. Therefore, any isomorphism on the 1-skeleton maps induces an isomorphism on the full space.

Step 3: Part (a) - Isomorphism of covering spaces

Covering spaces \(\tilde{X}_1 \rightarrow X\) and \(\tilde{X}_2 \rightarrow X\) are isomorphic if their 1-skeleton maps \(\tilde{X}_1^1 \rightarrow X^1\) and \(\tilde{X}_2^1 \rightarrow X^1\) are isomorphic. Since the covering maps act homeomorphically on cells, an isomorphism on the 1-skeleton extends to the entire covering.

Step 4: Definitions for Normal Covering Space

A covering space \(\tilde{X} \rightarrow X\) is normal if the group of deck transformations acts transitively on fibers above any point, which is also a property checked via its 1-skeleton by considerations of its action on loops.

Step 5: Part (b) - Normal Covering Space Condition

The restriction to the 1-skeleton retains normality since transitive action on loops/fibers in 1-skeleton implies this action on the entire complex. \(\tilde{X} \rightarrow X\) is normal iff \(\tilde{X}^1 \rightarrow X^1\) is normal.

Step 6: Define Deck Transformation Group

Deck transformations are automorphisms of the covering space that project to the identity on the base space. They are bijections at the level of covering maps that maintain the covering structure.

Step 7: Part (c) - Isomorphism of Deck Transformation Groups

Since deck transformations are determined by their action on the 1-skeleton, this means a deck transformation of the total covering induces one on the restricted covering. Therefore, the groups of deck transformations are isomorphic via restriction to 1-skeleton.

Key Concepts

CW complexes

CW complexes are a pivotal concept in topology, serving as a versatile framework for building spaces from simple pieces called cells. A CW complex consists of cells of diverse dimensions, each glued together in a specific, organized manner. This structure makes it particularly useful in both algebraic and geometric topology.

CW complexes are constructed using a sequence of steps:

• Start with a 0-dimensional space, usually a discrete set of points called the 0-skeleton.
• Attach 1-dimensional cells (think of lines or circles), resulting in a 1-skeleton.
• Progressively attach higher-dimensional cells to the existing structure, ensuring each attachment is coherent with the previous level.

The beauty of CW complexes lies in their flexibility. Despite being simple structures, they manage to model very complex spaces. This makes them incredibly useful for studying topological properties of spaces because calculations are easier on such decomposed structures. The attachment of cells ensures that any topology derived from CW complexes is manageable and can be broken down into comprehensible parts for both visualization and mathematical manipulation.

deck transformations

Deck transformations are an intriguing aspect of covering spaces. They are the set of all homeomorphisms from a covering space to itself, showcasing a unique property: they must project down to the identity map on the base space. In essence, deck transformations represent the symmetries of the covering structure that preserve its form.

Key attributes of deck transformations include:

• They are automorphisms, meaning they are bijective, structure-preserving maps from the covering space onto itself.
• A deck transformation must map fibers above a point in the base space to other fibers above the same point, essentially rearranging these fibers.
• The set of all such transformations forms a group, indicating operations like combination and inversion are defined among these transformations.

Understanding deck transformations is crucial as they offer insight into the geometry and symmetry of the space. They can reveal whether a covering space is normal, meaning the group of these transformations acts transitively on the fibers. This characteristic greatly influences the complexity and classification of the space.

normal covering space

A normal covering space is a specific type of covering space with a special property: the group of deck transformations acts transitively on the fibers over any point in the base space. This means you can move from any point above the base to any other point through the action of a deck transformation.

Some features of normal covering spaces include:

• They reflect a form of symmetry across their fibers due to the transitive action of deck transformations.
• Normal covering spaces are associated with subgroups of the fundamental group of the base space.
• Given a normal covering, any loop in the base space can be lifted to a path that closes to a loop in the covering space.

Normal covering spaces are an essential concept when exploring the interplay between algebra and topology. They allow us to gain a deeper understanding of the fundamental group and its relation to covering spaces, shedding light on how these structures unfold in the broader context of topological spaces.