Problem 40

Question

From the long exact sequence of homology groups associated to the short exact sequence of chain complexes $0 \rightarrow C_{i}(X) \stackrel{n}{\longrightarrow} C_{i}(X) \rightarrow C_{i}\left(X ; \mathbb{Z}_{n}\right) \rightarrow 0$ deduce immediately that there are short exact sequences $$ 0 \rightarrow H_{i}(X) / n H_{i}(X) \rightarrow H_{i}\left(X ; Z_{n}\right) \rightarrow n-\text {Torsion}\left(H_{i-1}(X)\right) \rightarrow 0 $$ where $n$ -Torsion(G) is the kernel of the map $G \stackrel{n}{\longrightarrow} G, g \mapsto n g .$ Use this to show that $\tilde{H}_{i}\left(X ; \mathbb{Z}_{p}\right)=0$ for all $i$ and all primes $p$ iff $\tilde{H}_{i}(X)$ is a vector space over $\mathbb{Q}$ for all $i$.

Step-by-Step Solution

Verified

Answer

$\tilde{H}_{i}(X; \mathbb{Z}_{p}) = 0$ for all $i$ and primes $p$ if and only if $\tilde{H}_{i}(X)$ is a vector space over $\mathbb{Q}$ for all $i$.

1Step 1: Understand the Short Exact Sequence

We start with the given short exact sequence of chain complexes:\[ 0 \rightarrow C_{i}(X) \xrightarrow{n} C_{i}(X) \rightarrow C_{i}(X; \mathbb{Z}_{n}) \rightarrow 0 \]Here, each map is a homomorphism of chain complexes, and the image of one map is the kernel of the next. This setting allows us to use the long exact sequence in homology.

2Step 2: Applying the Long Exact Sequence in Homology

From the short exact sequence of chain complexes, we derive a long exact sequence of homology groups:\[\cdots \rightarrow H_{i}(X) \xrightarrow{n} H_{i}(X) \rightarrow H_{i}(X; \mathbb{Z}_{n}) \rightarrow H_{i-1}(X) \xrightarrow{n} H_{i-1}(X) \rightarrow \cdots\]Examining this sequence allows us to break the sequence into the parts necessary to form short exact sequences.

3Step 3: Forming the Short Exact Sequences

From the long exact sequence, we identify important sequences:\[ 0 \rightarrow \frac{H_{i}(X)}{nH_{i}(X)} \rightarrow H_{i}(X; \mathbb{Z}_{n}) \rightarrow \ker(n: H_{i-1}(X) \rightarrow H_{i-1}(X)) \rightarrow 0 \]Here, $n$-Torsion$H_{i-1}(X)$ is defined as $\ker(n: H_{i-1}(X) \rightarrow H_{i-1}(X))$, which gives us the short exact sequence.

4Step 4: Implications for Torsion-Free Case

Assume $\tilde{H}_{i}(X; \mathbb{Z}_{p}) = 0$ for all $i$ and primes $p$. This implies:1. $ H_{i}(X) / p H_{i}(X) = 0 $, indicating $H_{i}(X)$ is torsion-free for all primes $p$.2. If $H_{i}(X)$ is free of torsion, it must be a vector space over $\mathbb{Q}$.

5Step 5: Converse Implication

If $\tilde{H}_{i}(X)$ is a vector space over $\mathbb{Q}$, it is torsion-free:1. Therefore, for any prime $p$, we have $pH_{i}(X) = H_{i}(X)$, making $H_{i}(X) / pH_{i}(X) = 0$.2. Hence, $\tilde{H}_{i}(X; \mathbb{Z}_{p}) = 0$ for all $i$.

Key Concepts

Homology GroupsChain ComplexesExact SequenceTorsion-FreeVector Space over $\mathbb{Q}$

Homology Groups

Homology groups are an essential concept in algebraic topology. They measure the number of different types of cycles within a topological space. In simpler terms, cycles can be thought of as holes or voids, such as loops or cavities. Homology groups are used to categorize these cycles across different dimensions.
Homology groups are usually denoted as $H_i(X)$, with $i$ denoting the dimension. For example, $H_0(X)$ represents the group of connected components, while $H_1(X)$ relates to loops or circles. Understanding these groups can provide deep insights into the shape and connectivity of the space you're studying. They are the building blocks for defining more complex structures, such as exact sequences and chain complexes.

Chain Complexes

Chain complexes play a pivotal role in calculating homology groups. They are sequences of abelian groups (like modules or vector spaces) connected by homomorphisms with a composition that equals zero. This means if you go two steps in the sequence, you end up with zero.
A chain complex typically looks like \[ \cdots \rightarrow C_{i+1}(X) \rightarrow C_{i}(X) \rightarrow C_{i-1}(X) \rightarrow \cdots \]. The maps between them, called boundary operators, are essential. They allow you to break down complex spaces into simpler pieces to analyze each dimension's cycle and boundary.

These structures accommodate the calculation of homology groups.
Understanding chain complexes is a step towards accurately utilizing the long exact sequence of homology.

Exact Sequence

An exact sequence is an arrangement of algebraic structures and homomorphisms such that the image of one homomorphism matches the kernel of the following one. This property is crucial for chain and cochain complexes.
A sequence is called 'exact' if for any given section of it $... \rightarrow A \rightarrow B \rightarrow C \rightarrow ...$, the image of $A \rightarrow B$ is exactly the kernel of $B \rightarrow C$. This means there's no loss of information as you move through the sequence.
Exact sequences are significant in algebraic topology because they capture how structures like cycles and boundaries are interrelated. They also help in transitioning between homology groups, providing a structured way to study topological spaces.

Torsion-Free

In the context of algebraic topology, a group is deemed torsion-free if multiplying any non-zero element by a non-zero integer never results in zero. In other words, when the group has no elements with finite order (except the identity).
Torsion-free modules and groups are particularly nice because they resemble vector spaces more closely than their torsion-filled counterparts. This is important because:

They allow the homology groups to more freely represent cycles without being collapsed by torsion elements.
A torsion-free homology group can often be seen as a vector space over the rationals $\mathbb{Q}$.

Understanding whether a group is torsion-free helps in recognizing when certain simplifications and assumptions can be made about the topological space.

Vector Space over $\mathbb{Q}$

A vector space over $\mathbb{Q}$ means you have a set of elements for which you can perform vector addition and scalar multiplication, where the scalars come from the rational numbers $\mathbb{Q}$.
The key properties of vector spaces are:

Closure under addition and scalar multiplication.
Existence of a zero vector (an element that doesn't change other elements when added to them).
Every element has an additive inverse.
Distributive and associative laws apply.

When homology groups are vector spaces over $\mathbb{Q}$, it often implies they're torsion-free. This makes working with them convenient, as their structure aligns with our classic understanding of vector spaces in linear algebra.

Problem 37

Problem 41