Long exact sequence of a pair

Depends on

• Relative singular homology
• The long exact sequence in homology

Used by

• Euler number ±1 makes the Milnor sphere bundle a homotopy seven-sphere Corollary
• Good pairs and quotient reduced homology Corollary
• Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
• A homology cobordism need not be an h-cobordism Counterexample
• A homology equivalence need not be a homotopy equivalence without simple connectivity Counterexample
• Excision fails without closure inside interior Counterexample
• Absolute and relative Hurewicz homomorphisms Definition
• Cellular boundary from three consecutive skeleta Definition
• Local degree at an isolated preimage Definition
• Relative connecting homomorphism on cycles Definition
• The unreduced Burau relative homology module Definition
• A simply connected CW homology equivalence is a homotopy equivalence under the stated choice conditions Example
• Integral cohomology ring of a closed orientable surface Example
• Local degrees of a polynomial map on the riemann sphere Example
• Poincaré–Lefschetz duality for a disk Example
• Relative homology of a disk and its boundary Example
• Relative homology of an interval and its endpoints Example
• A collar constructs the relative orientation class and its boundary class Lemma
• A CW quotient induces relative singular homology isomorphisms Lemma
• A multiple of a fork surface has a closed compact replacement Lemma
• A product collar deformation retracts onto its face Lemma
• A rational homology four-ball has square boundary torsion order Lemma
• A relative single cell layer has compatible homotopy and homology bases Lemma
• Attaching handles of index at least q preserves homology below q-1 Lemma
• Cap duality on a Euclidean coordinate ball Lemma
• Cellular boundary coefficients are the signed trajectory counts Lemma
• Cellular reduction for a highly connected pair Lemma
• Compatible orientation classes over compact subsets Lemma
• Connected sum descends to oriented h-cobordism classes Lemma
• Connected sum preserves oriented homotopy spheres Lemma
• Coordinate-ball classes identify local homology stalks Lemma
• Integral homology of a wedge of higher spheres has its cell basis Lemma
• Integral surface cup pairing from the oriented polygon Lemma
• Local sphere orientations and finite puncture excision Lemma
• One handle changes relative homology in one degree only Lemma
• Orientation reversal is the connected-sum inverse Lemma
• Relative homology of an h-cobordism vanishes at both ends Lemma
• Relative homology of the standard handle pair Lemma
• Relative homology over one base cell is shifted fiber homology Lemma
• Serre-fibration replacement preserves fiber homology transport Lemma

…and 36 more results.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Allen Hatcher, Algebraic Topology, Theorem 2.16 (standard reference, not scraped)