Problem 7 - Algebraic Topology

Question

For a map \(f: M \rightarrow N\) between connected closed orientable \(n\) -manifolds with fundamental classes \([M]\) and \([N],\) the degree of \(f\) is defined to be the integer \(d\) such that \(f_{*}([M])=d[N],\) so the sign of the degree depends on the choice of fundamental classes. Show that for any connected closed orientable \(n\) -manifold \(M\) there is a degree \(1 \operatorname{map} M \rightarrow S^{n}.\)

Step-by-Step Solution

Answer

There is a degree 1 map from any closed orientable manifold to the sphere due to homotopy and mapping properties.

Step 1: Understand the Degree of a Map

The degree of a map, denoted as \(d\), between two manifolds \(M\) and \(N\) is an integer that represents the number of times the domain manifold \(M\) wraps over the range manifold \(N\). Specifically, for a map \(f: M \rightarrow N\), it is defined such that \(f_{*}([M]) = d[N]\) in terms of the fundamental classes \([M]\) and \([N]\) of \(M\) and \(N\) respectively.

Step 2: Recognize Orientability and Fundamental Class

Since \(M\) is a connected closed orientable \(n\)-manifold, it admits a fundamental class \([M]\) in singular homology \(H_n(M; \mathbb{Z})\). Similarly, the sphere \(S^n\) is also orientable and closed, admitting a fundamental class \([S^n]\) in \(H_n(S^n; \mathbb{Z})\).

Key Concepts

Degree of a Map

In algebraic topology, the concept of the degree of a map provides insight into how one manifold covers another. Given a continuous map \( f: M \rightarrow N \) between connected closed orientable \( n \)-manifolds, the degree, denoted \( d \), is an integer that tells us how \( M \) wraps around \( N \). The degree is determined by relating the fundamental classes \([M]\) and \([N]\) of the manifolds through the map induced on homology, such that \( f_{*}([M]) = d[N] \). This relationship essentially counts how many times the entire manifold \( M \) maps onto \( N \) in a way that respects their orientations. If \( d \) is positive, the orientations are preserved, while a negative \( d \) indicates a reversal. Understanding this degree helps in classifying and comparing different maps between manifolds.

Connected Manifolds

Connected manifolds play a crucial role when analyzing maps between spaces in topology. A manifold is connected if it is in a single piece, meaning there is a path connecting any two points within it. For instance, consider the manifold \( M \). If it is connected, it means there are no separate, disjoint parts within \( M \). This property is vital because maps between connected manifolds tend to have well-defined, global characteristics, such as the degree of the map. Here are some key points about connected manifolds:

• Ensures that continuity properties of maps hold over the entire space.
• Aid in understanding the topological properties across the whole manifold.
• Simplify the classification of maps, as each map affects the entire manifold uniformly.

Orientable Manifolds

Orientability is a property of manifolds that indicates the possibility of consistently defining a 'handedness' at every point. If a manifold is orientable, you can smoothly assign directions to it. This is crucial when considering maps between manifolds, especially when discussing degrees. An orientable manifold allows for the existence of a fundamental class in its homology, such as \([M]\) for manifold \( M \). Why does orientability matter?

• It ensures coherency in how volumes and directions are measured across the manifold.
• Lets you define a fundamental class, which is essential for talking about the degree of a map.
• Orientability affects how transformation maps behave, making certain topological operations feasible.

Fundamental Classes

In the realm of algebraic topology, fundamental classes are crucial as they represent the "whole" of a manifold in its top homology group. For a closed orientable \( n \)-manifold \( M \), the fundamental class is an element \([M]\) in \( H_n(M; \mathbb{Z}) \), the top dimension homology group. The fundamental class encapsulates the manifold's entire homological structure, allowing comparisons through maps. This is essential when considering the degree of a map, as it leverages these classes: \( f_{*}([M]) = d[N] \). Core aspects of fundamental classes include:

• Providing a means to calculate the degree of maps between manifolds.
• Representing the overall topology and "shape" of a manifold in homology.
• Being central to the classification and understanding of manifolds in topological terms.