Problem 18 - Algebraic Topology

Question

For the closed orientable surface \(M\) of genus \(g \geq 1,\) show that for each nonzero \(\alpha \in H^{1}(M ; \mathbb{Z})\) there exists \(\beta \in H^{1}(M ; \mathbb{Z})\) with \(\alpha \beta \neq 0 .\) Deduce that \(M\) is not homotopy equivalent to a wedge sum \(X \vee Y\) of \(C W\) complexes with nontrivial reduced homology. Do the same for closed nonorientable surfaces using cohomology with \(\mathbb{Z}_{2}\) coefficients.

Step-by-Step Solution

Answer

For closed surfaces, \(M\) cannot be decomposed as a wedge sum of smaller complexes with nontrivial homology due to non-degenerate cohomology products and pairing properties.

Step 1: Understanding cohomology classes for oriented surfaces

For a closed orientable surface of genus \(g\), consider the 1-dimensional cohomology group \(H^1(M; \mathbb{Z})\). For such a surface, this group is isomorphic to \(\mathbb{Z}^{2g}\), reflecting the fact that we can choose 2g generators that correspond to the loop generators of the fundamental group (essentially, the 'handles' of the surface).

Step 2: Product in cohomology

In cohomology, the cup product \(\cup\) maps \(H^1(M; \mathbb{Z}) \otimes H^1(M; \mathbb{Z}) \to H^2(M; \mathbb{Z})\). For a non-zero \(\alpha \, \in \, H^1(M; \mathbb{Z})\), we need another class \(\beta\) so that the product \(\alpha \cup \beta eq 0\).

Step 3: Existence of nonnull class \(\beta\)

Given that \(H^2(M; \mathbb{Z}) \equiv \mathbb{Z}\), a nonzero cohomology product \(\alpha \cup \beta\) can be a generator of \(H^2(M; \mathbb{Z})\). For closed orientable surfaces, the intersection form is non-degenerate, so there exists \(\beta\) such that \(\alpha \cup \beta eq 0\), essentially making \(M\) different from a wedge sum of lower-dimensional complexes.

Step 4: Non-homotopy equivalence implication

If \(M\) were homotopy equivalent to a wedge sum \(X \vee Y\), where each has nontrivial cohomology, then any cohomology class product \(\alpha \cup \beta\) would be zero unless \(\alpha\) and \(\beta\) each belong to different factors of the wedge sum. This contradicts the existence of \(\alpha, \beta\) with \(\alpha \cup \beta eq 0\), hence \(M\) is not homotopy equivalent to such a wedge sum.

Step 5: Nonorientable surfaces

For nonorientable surfaces, take cohomology with \(\mathbb{Z}_2\) coefficients. The same logic applies: for a closed nonorientable surface, \(H^1(M; \mathbb{Z}_2)\) is non-zero, and using the non-degenerate pairing property, there exists \(\alpha, \beta\) such that \(\alpha \cup \beta eq 0\). Thus, similar to the orientable case, \(M\) cannot be decomposed into smaller homotopy components \(X \vee Y\) with nontrivial reduced homology.

Key Concepts

Orientable Surfaces

A surface is called orientable if you can define a consistent "direction" or orientation across the entire surface. Imagine walking around on the surface and always knowing which side is "up"—that's what being orientable means.
For closed surfaces, such as the sphere or the torus, orientations can be easily imagined. But for more complicated surfaces, such as a two-holed torus (genus 2), consistent orientation becomes more nuanced.

• Closed Orientable Surfaces: These are surfaces like the torus, where no cutting is needed to complete the surface.
• Genus of a Surface: The number of "holes" or "handles" that the surface has, like the single hole in a standard torus.

Understanding orientable surfaces is crucial as it helps determine properties in topology, especially in the study of cohomology.

Cup Product

The cup product is an operation used in algebraic topology, specifically within cohomology. It helps us combine classes from cohomology groups to get new ones.
For a surface, this multiplication works by combining classes from the first cohomology group, which typically represent non-trivial loops or cycles on the surface. The cup product

• Takes Two Classes: Consider two 1-forms or cohomology classes from the group like loops.
• Combines into a New Class: Resulting in a class in the second cohomology group for the same surface. This is like creating a "covering" over both loops.

This is important because a non-zero product implies additional structures and properties of the surface.

Genus

The genus of a surface is a fundamental topological property that essentially counts the number of "holes" or "handles" a surface has. It's denoted by the letter \(g\). The term was popularized in part to generalize the concepts related to the geometry of surfaces.

• Genus 0: A sphere with no holes.
• Genus 1: A torus, which looks like a doughnut with a single hole.
• Genus 2: Like two toruses connected together or a surface with two holes.

The genus helps us classify surfaces and understand their complex topological properties, including intrinsic characteristics like cohomology groups.

Wedge Sum

A wedge sum is a way to glue multiple topological spaces together at a single point. It is denoted by \(X \vee Y\).
Think of taking two surfaces, poking a hole in each, and then connecting them together at one point, making them share that point.

• Simple Construction: Formed by identifying a single point from each space.
• Topological Impact: The combined space inherits properties from both spaces.

In topology, understanding wedge sums provides crucial insights into whether a space can be decomposed into simpler components or if it's irreducibly complex.

Homotopy Equivalence

Homotopy equivalence is a fundamental concept that groups spaces into equivalence classes, where two spaces are considered the same for topological purposes if they can be continuously deformed into one another.
This concept shines in understanding complex topological structures. If two spaces are homotopy equivalent, they share many topological properties.

• Deformation: It focuses on whether one space can be transformed into another smoothly, without tearing or gluing.
• Topological Classification: Helps classify spaces by their intrinsic properties, rather than their exact physical shape.

An intricate property of surfaces is sealed within homotopy equivalence. With complex surfaces, this reveals whether they can or cannot be expressed as simpler compositions of other spaces.