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The rational Hurewicz range for the four-sphere
Example
Assume AC. For S⁴, the rational Hurewicz map is an isomorphism in degrees 4, 5, and 6: π₄(S⁴)⊗Q≅H₄(S⁴;Q)≅Q, and both π_i(S⁴)⊗Q and H_i(S⁴;Q) vanish for i=5,6.
Facts & Assumptions
Given: AC; the standard based CW structure on with one -cell and one -cell; the cell-pushing lemma for low-dimensional disks; and the sphere homology computation with coefficients in .
The low-dimensional disk-pushing lemma deforms a based cube map of dimension into the 0-cell while fixing its boundary, so is 3-connected (A low-dimensional disk can be pushed off a higher cell).
The sphere homology computation with coefficient group gives the displayed rational homology groups (Homology of spheres), and the rational Hurewicz theorem applies with in the range , namely (Rational Hurewicz for highly connected CW complexes).
The rational sphere homotopy computation identifies in that range (Rational sphere homotopy below the first unstable degree), and AC is inherited from the rational Hurewicz theorem (The Axiom of Choice).
Verification
Give S⁴ its standard based CW structure with one 0-cell and one 4-cell. For each j=1,2,3, a based cubical representative f:Iʲ→S⁴ has boundary mapped to the 0-cell. Apply lem-a-low-dimensional-disk-can-be-pushed-off-a-higher-cell to the finite one-cell attachment (S⁴,*), with n=j<4. It deforms f into the 0-cell while fixing its boundary, so π₁(S⁴)=π₂(S⁴)=π₃(S⁴)=0. This writes out the one-cell connectivity argument using the published cell-pushing supplier; the stronger lem-high-relative-cells-do-not-change-lower-homotopy is also available in page 547's published prerequisite closure but is not needed as a direct dependency. cor-homology-of-spheres with coefficient group Q gives the displayed rational homology groups directly.
Now apply the rational Hurewicz theorem with c=4; its range is c≤i≤2c−2, namely 4≤i≤6. The rational sphere lemma computes the three homotopy groups. The degree-4 map is the first-nonzero-degree Hurewicz isomorphism; in degrees 5 and 6 both sides vanish.
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Sources
- Daniel S. Freed, Bordism: Old and New (standard reference, not scraped)
- Allen Hatcher, Algebraic Topology (standard reference, not scraped)