The excision theorem says H_n(X \ Z, A \ Z) ≅ H_n(X, A) provided that the closure of Z is contained in the interior of A. Why is this condition necessary?
AIt ensures that Z does not touch the boundary of A, so removing Z does not change the topology near where A meets X \ A
BIt ensures that X \ Z is still a topological space
CIt ensures the chain groups remain finitely generated
DIt ensures that A \ Z is contractible
The condition cl(Z) ⊂ int(A) ensures that Z is 'buried deep inside A,' far from the boundary where A meets its complement in X. Relative homology H_n(X, A) is sensitive to the topology near the boundary of A in X — it measures what happens in X that does not already happen in A. Removing Z from deep inside A changes neither the space near this boundary nor the relative chains that detect it. Without this condition, Z might intersect the boundary of A, genuinely altering the relative topology.
Question 2 True / False
Excision implies that for a good pair (X, A), the relative homology H_n(X, A) is isomorphic to the reduced homology H̃_n(X/A).
TTrue
FFalse
Answer: True
For good pairs (where A is a neighborhood deformation retract), excision and the long exact sequence combine to show H_n(X, A) ≅ H̃_n(X/A). Intuitively: collapsing A to a point is the geometric version of 'modding out by A,' and relative homology is the algebraic version. Excision provides the key step by showing that the homology only sees the local behavior near the boundary of A. This result is foundational for cellular homology, where H_n(X^n, X^{n-1}) ≅ H̃_n(X^n/X^{n-1}) ≅ H̃_n(∨S^n) detects the n-cells.
Question 3 Multiple Choice
In the proof of the Mayer-Vietoris sequence, excision plays which role?
AIt shows the boundary maps are zero
BIt identifies the relative homology H_n(X, A) with H_n(B, A ∩ B), connecting the homology of the pieces
CIt proves that the chain groups are free abelian
DIt establishes functoriality of homology
The Mayer-Vietoris sequence for X = A ∪ B is derived from the long exact sequence of the pair (X, A). Excision provides the crucial identification: H_n(X, A) ≅ H_n(B, A ∩ B) (excising Z = X \ B from the pair (X, A), noting that cl(X \ B) ⊆ int(A) when {int(A), int(B)} cover X). This identification converts the long exact sequence of (X, A) into the Mayer-Vietoris sequence, which involves only the homology groups of A, B, and A ∩ B.
Question 4 Short Answer
Explain why excision makes homology 'local' in a way that homotopy groups are not.
Think about your answer, then reveal below.
Model answer: Excision says that relative homology H_n(X, A) is insensitive to changes deep inside A or far from A — it depends only on the local topology near the boundary where A meets X \ A. This is a locality property that homotopy groups do not share: π_n(X, A) does not satisfy excision in general (the failure of excision for homotopy groups is related to the Freudenthal suspension theorem and is measured by higher connectivity conditions). Homology's local character is what makes it computationally powerful — we can decompose spaces into pieces and compute the homology of each piece independently.
The difference is fundamental. Homology is an 'abelian' invariant (the chain groups are abelian), and abelianization introduces the locality that excision captures. Homotopy groups, being non-abelian in general, carry global information that cannot be localized. This is why homology is easier to compute (excision + Mayer-Vietoris give systematic decomposition methods) while homotopy groups resist computation (the homotopy groups of spheres remain only partially known).