physics

Relative homology

Relative homology is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Relative homology rather than just read about it. In short: In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways.

Key takeaways

Relative homology belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Relative homology to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Relative homology from memory before moving on to harder problems.

Reference excerpt

In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

Definition Given a subspace A ⊆ X {\displaystyle A\subseteq X} , one may form the short exact sequence

0 → C ∙ ( A ) → C ∙ ( X ) → C ∙ ( X ) / C ∙ ( A ) → 0 , {\displaystyle 0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)\to 0,}

where C ∙ ( X ) {\displaystyle C_{\bullet }(X)} denotes the singular chains on the space X. The boundary map on C ∙ ( X ) {\displaystyle C_{\bullet }(X)} descendsa to C ∙ ( A ) {\displaystyle C_{\bullet }(A)} and therefore induces a boundary map ∂ ∙ ′ {\displaystyle \partial '_{\bullet }} on the quotient. If we denote this quotient by C n ( X , A ) := C n ( X ) / C n ( A ) {\displaystyle C_{n}(X,A):=C_{n}(X)/C_{n}(A)} , we then have a complex

⋯ ⟶ C n ( X , A ) → ∂ n ′ C n − 1 ( X , A ) ⟶ ⋯ . {\displaystyle \cdots \longrightarrow C_{n}(X,A)\xrightarrow {\partial '_{n}} C_{n-1}(X,A)\longrightarrow \cdots .}

By definition, the nth relative homology group of the pair of spaces ( X , A ) {\displaystyle (X,A)} is

H n ( X , A ) := ker ⁡ ∂ n ′ / im ⁡ ∂ n + 1 ′ . {\displaystyle H_{n}(X,A):=\ker \partial '_{n}/\operatorname {im} \partial '_{n+1}.}

One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e., chains that would be boundaries, modulo A again).

Properties The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence

⋯ → H n ( A ) → i ∗ H n ( X ) → j ∗ H n ( X , A ) → ∂ H n − 1 ( A ) → ⋯ . {\displaystyle \cdots \to H_{n}(A){\stackrel {i_{*}}{\to }}H_{n}(X){\stackrel {j_{*}}{\to }}H_{n}(X,A){\stackrel {\partial }{\to }}H_{n-1}(A)\to \cdots .}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Relative homology

Start with the simplest possible case. Write down what Relative homology claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Relative homology before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Relative homology ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Relative homology

In research: Relative homology appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Relative homology in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Relative homology is common in secondary-school and first-year university syllabi. It links to neighbouring topics Homology theory, so understanding it makes those chapters shorter.
In everyday life: Look for Relative homology outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Relative homology in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Relative homology means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Relative homology out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Relative homology in simple terms?

Why does Relative homology matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Relative homology?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Relative homology.