03.08.20 · modern-geometry / k-theory

Whitehead torsion and the s-cobordism theorem

shipped3 tiersLean: none

Anchor (Master): Milnor *Whitehead torsion* (Bull. AMS 72, 1966) full; Whitehead 1950 (*Ann. of Math.* 52); Kervaire 1965 (*Comment. Math. Helv.* 40, s-cobordism); Cohen *A Course in Simple-Homotopy Theory*; Lück *A Basic Introduction to Surgery Theory* §1

Intuition Beginner

Two shapes can be the same in a loose sense — you can deform one into the other without tearing — and yet differ in a sharper, more honest sense. Imagine building a shape out of soft blocks. Sometimes you can simplify it by collapsing a block that is glued on along a free face: push it flat and it disappears, like deflating a bump. The reverse move grows a block back. Two shapes are simply equivalent when one turns into the other by these honest collapse-and-grow moves alone.

The surprise is that some shapes are loosely the same but are not simply equivalent. There is a hidden obstruction, a kind of leftover bookkeeping, that the collapse moves can never clear away. This leftover is a single algebraic stamp attached to the deformation. When the stamp reads zero, the two shapes really are buildable from each other by honest moves. When it does not, they are stuck apart in this finer sense even though they looked the same.

This stamp is the Whitehead torsion, and it lives in a group built from the symmetries of the shape's loops.

Visual Beginner

Picture a strip of paper with a small flap glued along one of its edges. Because the flap is attached only along that one edge, you can fold it flat onto the strip and it vanishes — the strip is left simpler but unchanged in shape. That is a collapse. Run it backwards and the flap grows out again — an expansion. A sequence of these moves is the only kind of change that counts as "honest."

The takeaway: collapses and expansions are the honest simplifying moves. The wavy arrow is a looser deformation. The little tag $τ$ is the leftover stamp that measures whether the loose deformation can be replaced by honest moves. Zero stamp means yes.

Worked example Beginner

A leftover stamp from a $2$ -by- $2$ table. The stamp lives in a bookkeeping system built from square tables of numbers that can be lined up by simple row moves. Here is the flavour of when a table counts as "clearable."

Take the table $$ \begin{pmatrix} 1 & 5 \ 0 & 1 \end{pmatrix}. $$ You can clear it back to the plain table $(1001)$ by one allowed move: subtract $5$ times the second row from the first. After that single shear the off-diagonal $5$ is gone and the table is plain. A table you can reduce to the plain one by such shears carries the zero stamp.

Now compare the table $(3001)$ , which scales one direction by $3$ . Shears never change the product down the diagonal, so this scaling cannot be cleared to the plain table by shears alone. Its stamp records the leftover factor $3$ .

What this tells us. The honest moves correspond to shears — and these clear most tables to nothing. What survives is the scaling part, the determinant-like leftover. Whitehead torsion is exactly this surviving leftover, computed not from one table but from the whole stack of tables describing how one shape collapses onto another.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $π$ is a group and $Z π$ its integral group ring. Modules are left $Z π$ -modules; "based free" means free with a chosen ordered basis.

Definition (the Whitehead group). Let $GL (Z π) = lim_{n} GL_{n} (Z π)$ be the stable general linear group, with $A \mapsto \begin{psmallmatrix} A & 0 \\ 0 & 1 \end{psmallmatrix}$ the stabilisation. Let $E (Z π) ⊴ GL (Z π)$ be the subgroup generated by elementary matrices (identity plus a single off-diagonal entry). The Whitehead lemma identifies $E (Z π) = [GL (Z π), GL (Z π)]$ , so the quotient $$ K_1(\mathbb{Z}\pi) = \mathrm{GL}(\mathbb{Z}\pi)/E(\mathbb{Z}\pi) $$ is abelian. Inside it sit the classes of the units $\pm π \subset (Z π)^{\times}$ , the $\pm π$ -units. The Whitehead group of $π$ is the further quotient $$ \mathrm{Wh}(\pi) = K_1(\mathbb{Z}\pi)\big/\langle \pm g : g \in \pi\rangle . $$ This is the receptacle for torsion: it is the part of $K_{1}$ that is invisible to relabelling a basis by signs and group elements. Notation here ( $GL$ , $E$ , $K_{1}$ , $Wh$ , $Z π$ ) is introduced in this section and used consistently below.

Definition (torsion of an acyclic based complex). Let $$ C_* : \quad 0 \to C_n \xrightarrow{\partial} C_{n-1} \xrightarrow{\partial} \cdots \xrightarrow{\partial} C_0 \to 0 $$ be a finite chain complex of based free $Z π$ -modules that is acyclic ( $H_{*} (C_{*}) = 0$ ). Acyclicity provides a chain contraction $δ : C_{*} \to C_{*+ 1}$ with $\partial δ + δ \partial = id$ . Then $\partial + δ$ is an isomorphism from $C_{odd} = ⨁ C_{2 i + 1}$ to $C_{even} = ⨁ C_{2 i}$ , and relative to the chosen bases it is a matrix $A \in GL (Z π)$ once the two sides are stably identified. Its class is independent of the contraction, and the torsion is $$ \tau(C_*) = [A] \in \mathrm{Wh}(\pi). $$ A based complex with $τ (C_{*}) = 0$ is called simple.

Definition (Whitehead torsion of a homotopy equivalence). Let $f : X \to Y$ be a homotopy equivalence of finite connected CW complexes with $π_{1} (Y) = π$ . Lift to universal covers, obtaining a $Z π$ -chain map $\tilde{f}_{#} : C_{*} (\tilde{X}) \to C_{*} (\tilde{Y})$ between the based (by lifted cells) cellular complexes. Its algebraic mapping cone $Cone (\tilde{f}_{#})$ is a based free $Z π$ -complex, and it is acyclic precisely because $f$ is a homotopy equivalence. The Whitehead torsion of $f$ is $$ \tau(f) = \tau\big(\mathrm{Cone}(\tilde f_#)\big) \in \mathrm{Wh}(\pi). $$ A homotopy equivalence with $τ (f) = 0$ is a simple homotopy equivalence; equivalently (Whitehead 1950) $f$ is homotopic to a composite of elementary expansions and collapses.

A non-example clarifying the quotient: over $π = 1$ one has $Z π = Z$ , $K_{1} (Z) = {\pm 1}$ from the determinant, and the $\pm π$ -units already exhaust this, so $Wh (1) = 0$ . No simply-connected homotopy equivalence carries torsion — the reason the obstruction was invisible until non-simply-connected spaces were examined.

Counterexamples to common slips

Torsion is not the determinant. Over a noncommutative $Z π$ there is no determinant into $Z π$ ; $K_{1}$ is the correct replacement, and the Dieudonné determinant exists only over division rings.
Acyclicity is essential. The torsion is defined for acyclic based complexes. For a homotopy equivalence the cone is acyclic; for a mere map it need not be, and $τ$ is undefined.
The basis matters, but only up to $\pm π$ . Reordering cells, or relabelling a lift by $g \in π$ or by a sign, changes $A$ by an elementary or $\pm π$ -unit factor, hence not its class in $Wh (π)$ . Passing to $Wh$ rather than staying in $K_{1}$ is what makes $τ$ a topological invariant.
$τ (f)$ and $τ (f^{- 1})$ . Under a homotopy inverse the torsion negates up to the induced map: $τ (f^{- 1}) = - (f^{- 1})_{*} τ (f)$ . It is not in general zero.

Key theorem with proof Intermediate+

Theorem (s-cobordism theorem; Barden, Mazur, Stallings). Let $(W; V_{0}, V_{1})$ be a compact smooth h-cobordism with $dim W = m \geq 6$ , and write $π = π_{1} (V_{0})$ . The inclusion $V_{0} ↪ W$ is a homotopy equivalence, so it has a Whitehead torsion $τ (V_{0} ↪ W) \in Wh (π)$ . Then $W$ is diffeomorphic to the product $V_{0} \times [0, 1]$ if and only if $τ (V_{0} ↪ W) = 0$ . Moreover every element of $Wh (π)$ is realised as the torsion of some h-cobordism on $V_{0}$ , and $τ (V_{1} ↪ W) = (- 1)^{m - 1} \overline{τ (V_{0} ↪ W)}$ under the involution $\overline{(\cdot)}$ induced by $g \mapsto g^{- 1}$ .

Proof. Choose a self-indexing Morse function on $W$ relative to its ends, with gradient-like field, as developed in 03.02.21. Its handles give the universal cover $W$ a based free $Z π$ -cellular structure relative to $V_{0}$ , and the resulting chain complex $C_{*} (W, V_{0})$ is acyclic because $V_{0} ↪ W$ is a homotopy equivalence. By definition its torsion is $τ = τ (V_{0} ↪ W) \in Wh (π)$ , computed from the matrices of the boundary maps, whose entries are the $π$ -weighted intersection numbers of ascending and descending spheres in the cover ^{[fasttrack-texts gradient-like vector fields and the handle chain complex]}.

The geometric engine is handle manipulation. Three moves on a handle decomposition leave $W$ unchanged while changing the based complex by simple operations — operations that alter the torsion matrix only by elementary and $\pm π$ -unit factors: (i) handle slides, sliding one handle's attaching sphere across another of the same index, which add a unit-row multiple to a matrix and so realise elementary column operations over $Z π$ ; (ii) creation/cancellation of a complementary pair of adjacent-index handles meeting transversally in a single point, by 03.02.22, which stabilises or destabilises the matrix; (iii) reordering by the rearrangement of 03.02.21. Together these generate exactly the moves that do not change a class in $Wh (π)$ .

After arranging all handles in the middle indices and using the Whitney trick of 03.02.22 (available since $m \geq 6$ , so the middle level has dimension $m - 1 \geq 5$ ), the boundary matrix of $C_{*} (W, V_{0})$ between the two middle index groups is an element $A \in GL (Z π)$ whose class is $τ$ . If $τ = 0$ , then $A \in E (Z π) \cdot ⟨ \pm π ⟩$ : it is a product of elementary and $\pm π$ -unit matrices. Each elementary factor is realised geometrically by a handle slide, and each $\pm π$ -unit factor by a basis relabelling; performing these reduces $A$ to the identity, which means the attaching spheres of the higher-index handles meet the belt spheres of the lower-index handles in single points. The First Cancellation Theorem of 03.02.22 then cancels all handles in pairs, leaving a handle-free cobordism, i.e. $W ≅ V_{0} \times [0, 1]$ . Conversely, if $W ≅ V_{0} \times [0, 1]$ the inclusion is a simple homotopy equivalence (the product structure collapses), so $τ = 0$ . Realisation of arbitrary torsion is achieved by attaching a complementary pair of handles whose linking matrix is a prescribed $A \in GL (Z π)$ , and the duality formula follows from comparing the decompositions induced by $f$ and $1 - f$ . $■$

Setting $π = 1$ gives $Wh (1) = 0$ , so $τ$ vanishes automatically: every simply-connected h-cobordism of dimension $\geq 6$ is a product. This recovers the simply-connected h-cobordism theorem of 03.02.23.

Exercises Intermediate+

Exercise (hard, short-answer). Two h-cobordisms $W, W'$ on the same $V_0$ (with $\dim \ge 6$) have torsions $\tau, \tau' \in \mathrm{Wh}(\pi_1 V_0)$. Their "stacking" $W \cup_{V_1} W'$ is again an h-cobordism. Argue that its torsion is $\tau + \tau'$, and deduce when the stack is a product.

Hint

Torsion is additive over a composite cobordism (the based complexes concatenate), and the stack is a product iff its total torsion vanishes.

Answer

The handle decomposition of the stack is the concatenation of those of $W$ and $W^{'}$ , so the acyclic based complex is a (filtered) sum and torsion is additive: $τ (W \cup W^{'}) = τ + τ^{'}$ in $Wh (π_{1} V_{0})$ . By the s-cobordism theorem the stack is a product iff $τ + τ^{'} = 0$ , i.e. iff $τ^{'} = - τ$ . In particular gluing $W$ to its inverse cobordism (torsion $- τ$ ) yields a product — the geometric meaning of the group inverse.

Exercise (hard, short-answer). The lens space $L(7,1)$ and $L(7,2)$ are homotopy equivalent but not simple homotopy equivalent. Using $\mathrm{Wh}(\mathbb{Z}/7) \cong \mathbb{Z}$, explain in one or two sentences how Whitehead torsion distinguishes them, and what topological conclusion follows (no full computation required).

Hint

A homotopy equivalence $L (7, 1) \to L (7, 2)$ has a Reidemeister/Whitehead torsion in $Wh (Z /7)$ ; non-vanishing of that class obstructs simple equivalence and hence homeomorphism.

Answer

A homotopy equivalence $f : L (7, 1) \to L (7, 2)$ has torsion $τ (f) \in Wh (Z /7) ≅ Z$ , computable from the cellular chain complexes as a unit of $Z [Z /7]$ modulo the $\pm π$ -units (a Reidemeister-torsion calculation). For these parameters $τ (f) \neq = 0$ , so no homotopy equivalence between them is simple. Since a homeomorphism of finite CW complexes is a simple homotopy equivalence (Chapman's theorem), the two lens spaces are not homeomorphic, even though they are homotopy equivalent.

Lean formalization Intermediate+

Mathlib provides $K_{1}$ of a ring as a quotient of the stable general linear group but does not yet package the group-ring Whitehead group or chain-complex torsion, so this unit ships with lean_status: none. The statements one would target read, in schematic Lean 4 / Mathlib notation:

-- Whitehead group of a group π (target definition, not yet in Mathlib)
-- Wh π := K₁ (MonoidAlgebra ℤ π) ⧸ (Subgroup generated by ±π)
--
-- noncomputable def Whitehead (π : Type*) [Group π] : Type* :=
--   (K₁ (MonoidAlgebra ℤ π)) ⧸ pmPiUnits π
--
-- Torsion of a finite acyclic based complex of free ℤπ-modules:
-- def torsion {π} (C : ChainComplex (FreeModule (MonoidAlgebra ℤ π)) ℕ)
--     (hC : ChainComplex.Acyclic C) (b : Basis C) : Whitehead π := ...
--
-- Multiplicativity (the load-bearing lemma):
-- theorem torsion_comp (f g : HomotopyEquiv X Y) :
--     τ (g.comp f) = τ g + (π₁map g) (τ f)

Mathlib currently reaches Matrix.GeneralLinearGroup and a partial $K_{1}$ ; the gap is the group ring as a functor of $π$ , the contraction-matrix torsion of an acyclic based complex with its invariance lemmas, and the cellular mapping-cone construction. See the Mathlib gap analysis note for the itemised targets.

Advanced results Master

Reidemeister torsion and the abelianised invariant. When $π$ acts through a representation $ρ : π \to GL_{k} (F)$ over a field $F$ , the based $Z π$ -complex of a finite CW pair tensors to a based $F$ -complex $C_{*} \otimes_{ρ} F^{k}$ . If this is acyclic its $K_{1} (F) = F^{\times}$ -torsion is the Reidemeister torsion $Δ_{ρ} \in F^{\times} / (\pm det ρ (π))$ . It is the image of Whitehead torsion under $Wh (π) \to K_{1} (F)$ and predates it: Reidemeister, Franz, and de Rham used $Δ_{ρ}$ to classify the lens spaces $L (p, q)$ up to homeomorphism in the 1930s, decades before $Wh (π)$ was isolated. The lens-space computation runs entirely inside the cyclotomic ring $Z [ζ_{p}]$ : $Wh (Z / p)$ is free abelian of rank $(p - 3) /2$ for $p$ an odd prime, and the torsion of $L (p, q)$ is a unit of $Z [ζ_{p}]$ whose class detects $q mod p$ up to the equivalences $q \sim q^{- 1} \sim - q$ .

The Bass-Heller-Swan fundamental theorem. For a regular ring $R$ , $K_{1} (R [t, t^{- 1}]) ≅ K_{1} (R) \oplus K_{0} (R)$ . Applied to $R = Z$ and iterated, it yields $Wh (Z^{n}) = 0$ for all $n$ : free abelian fundamental groups carry no torsion, so an h-cobordism over a torus is automatically a product. The theorem also produces the Nil and lower $K$ -theory ( $K_{- n}$ ) terms that govern Whitehead groups of infinite groups, the input to the Farrell-Jones programme.

Structure of $Wh (π)$ for finite $π$ . For finite $π$ , $Wh (π)$ is finitely generated of rank $r - q$ , where $r$ is the number of irreducible real representations and $q$ the number of irreducible rational ones (the rank of $K_{1} (Z π)$ counted by the Hasse-Schilling-Maass reduced norm). Its torsion subgroup $SK_{1} (Z π)$ is finite and was computed in many cases by Oliver. The vanishing $Wh (π) = 0$ holds for $π$ free, free abelian, or the one-element group; it fails already for $π = Z /5$ , where $Wh (Z /5) ≅ Z$ .

Topological invariance and Chapman's theorem. A priori $τ (f)$ uses the CW (hence combinatorial) structure. Chapman's theorem (1974) removes this: a homeomorphism of finite CW complexes is a simple homotopy equivalence, so Whitehead torsion of a homeomorphism vanishes and simple homotopy type is a topological invariant. This is what licenses the conclusion that lens spaces with distinct Reidemeister torsion are non-homeomorphic.

Full proof set Master

$Wh (1) = 0$ . $Z \cdot 1 = Z$ is a Euclidean domain, so $GL (Z)$ is generated by elementary matrices and diagonal $\pm 1$ matrices; thus $K_{1} (Z) = Z^{\times} = {\pm 1}$ via the determinant. The $\pm π$ -units of $π = 1$ are $\pm 1$ , exhausting $K_{1}$ , so $Wh (1) = 0$ . $■$

Well-definedness of $τ (C_{*})$ . Given two chain contractions $δ, δ^{'}$ of an acyclic based complex $C_{*}$ , the maps $\partial + δ$ and $\partial + δ^{'}$ from $C_{odd}$ to $C_{even}$ differ by $id + (δ - δ^{'}) (\partial + δ)^{- 1}$ , where $(δ - δ^{'})$ raises degree, so $(δ - δ^{'}) (\partial + δ)^{- 1}$ is nilpotent and the correction is a product of elementary matrices. Hence $[\partial + δ] = [\partial + δ^{'}] \in K_{1}$ , and a fortiori in $Wh (π)$ . A change of ordered basis multiplies the matrix by a $\pm π$ -unit and elementary factor, again invisible in $Wh (π)$ . $■$

Additivity. For a short exact sequence $0 \to C_{*}^{'} \to C_{*} \to C_{*}^{''} \to 0$ of acyclic based complexes whose bases are compatible (the basis of $C_{*}$ is the concatenation of those of $C_{*}^{'}$ and $C_{*}^{''}$ ), one has $τ (C_{*}) = τ (C_{*}^{'}) + τ (C_{*}^{''})$ . The matrix of a contraction of $C_{*}$ is block-triangular over the contractions of the sub- and quotient complexes; its $K_{1}$ -class is the sum of the diagonal blocks' classes since the off-diagonal block contributes an elementary factor. $■$

The composition formula $τ (g \circ f) = τ (g) + g_{*} τ (f)$ follows from additivity applied to the cone sequence $0 \to Cone (\tilde{g}_{#}) \to Cone ((g f)_{#}) \to Σ Cone (\tilde{f}_{#})_{g} \to 0$ , where the third term is the cone of $f$ pushed forward along $g$ , contributing $g_{*} τ (f)$ , and the suspension flips no sign on torsion. $■$

The s-cobordism theorem itself is proved in the Intermediate section; the realisation and duality clauses are stated there with proofs sketched, and their full handle-theoretic detail is the content of 03.02.22–03.02.23.

Connections Master

03.02.22 (the Whitney trick and handle cancellation) supplies the geometric engine: every elementary and $\pm π$ -unit factor that the torsion class quotients away is realised by a handle slide or a single-point cancellation, and the Whitney trick in dimension $\geq 5$ is what lets an algebraically cancellable handle pair be cancelled geometrically. Whitehead torsion is precisely the obstruction that survives when that geometric cancellation is blocked by a larger fundamental group $π_{1} \neq = 1$ .
03.02.23 (the h-cobordism theorem) is the $π = 1$ specialisation: since $Wh (1) = 0$ , the torsion obstruction vanishes automatically and every simply-connected h-cobordism of dimension $\geq 6$ is a product. The s-cobordism theorem is its honest generalisation to arbitrary fundamental group, and it repays the simply-connected statement as a corollary.
03.08.01 (topological K-theory) is the $K_{0}$ / $K_{1}$ companion: where topological K-theory is the Grothendieck $K_{0}$ of vector bundles, Whitehead torsion is an algebraic $K_{1}$ invariant of the group ring $Z π$ . The same stabilise-then-quotient pattern — $GL / E$ versus virtual bundles — runs through both, and the long exact sequence relating $K_{0}$ and $K_{1}$ is the algebraic backbone shared with the Bass-Heller-Swan theorem.
03.12.20 (Whitehead's theorem) detects homotopy equivalences by isomorphism on $π_{n}$ ; Whitehead torsion refines this by measuring how far such an equivalence is from being built by elementary expansions and collapses, the finer "simple" relation Whitehead introduced in the same period.
07.02.03 (Grothendieck groups and the cde-triangle) develops the representation-theoretic $K_{0}$ of a finite group; the units of $Z π$ controlling $Wh (π)$ for finite $π$ are read off the same character theory (the Hasse-Schilling-Maass reduced norm), tying the rank of the Whitehead group to the count of rational versus real irreducibles.

Historical & philosophical context Master

J.H.C. Whitehead introduced simple homotopy type in Simple homotopy types (Amer. J. Math. 72, 1950), building on his 1939–1941 work on the combinatorial homotopy of complexes; the obstruction group $Wh (π)$ now bears his name. The invariant generalised the Reidemeister torsion of Reidemeister (1935), Franz (1935), and de Rham, who had used an abelianised version to distinguish lens spaces — the first invariant in topology to separate spaces that are homotopy equivalent but not homeomorphic.

Milnor's survey Whitehead torsion (Bull. AMS 72, 1966) consolidated the algebraic theory in the language of $K_{1}$ of group rings and made it standard. The geometric payoff, the s-cobordism theorem, was proved independently around 1963–1965 by Barden, Mazur, and Stallings; Kervaire's account (Comment. Math. Helv. 40, 1965) is the usual citation. The supporting algebraic K-theory was placed on a functorial footing by Bass, Heller, and Swan (Publ. Math. IHÉS 22, 1964), whose fundamental theorem computes the Whitehead groups of free abelian groups and seeds the lower K-theory used throughout surgery. Chapman (1974) proved the topological invariance that frees the torsion from its combinatorial definition. The s-cobordism theorem became the base case of the surgery classification of manifolds developed by Wall and Browder, where $Wh (π)$ and its $L$ -theoretic siblings index the obstructions.

Bibliography Master

Whitehead, J. H. C. Simple homotopy types. American Journal of Mathematics 72 (1950), 1–57.
Milnor, J. Whitehead torsion. Bulletin of the American Mathematical Society 72 (1966), 358–426.
Kervaire, M. Le théorème de Barden-Mazur-Stallings. Commentarii Mathematici Helvetici 40 (1965), 31–42.
Bass, H.; Heller, A.; Swan, R. The Whitehead group of a polynomial extension. Publications Mathématiques de l'IHÉS 22 (1964), 61–79.
Cohen, M. M. A Course in Simple-Homotopy Theory. Graduate Texts in Mathematics 10, Springer, 1973.
Lück, W. A Basic Introduction to Surgery Theory. ICTP Lecture Notes 9, 2002.
Oliver, R. Whitehead Groups of Finite Groups. London Mathematical Society Lecture Note Series 132, Cambridge University Press, 1988.
Chapman, T. A. Topological invariance of Whitehead torsion. American Journal of Mathematics 96 (1974), 488–497.

@article{Whitehead1950,
  author  = {Whitehead, J. H. C.},
  title   = {Simple homotopy types},
  journal = {American Journal of Mathematics},
  volume  = {72},
  pages   = {1--57},
  year    = {1950}
}
@article{Milnor1966torsion,
  author  = {Milnor, John},
  title   = {Whitehead torsion},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {72},
  pages   = {358--426},
  year    = {1966}
}
@article{Kervaire1965,
  author  = {Kervaire, Michel},
  title   = {Le th\'eor\`eme de Barden-Mazur-Stallings},
  journal = {Commentarii Mathematici Helvetici},
  volume  = {40},
  pages   = {31--42},
  year    = {1965}
}
@article{BassHellerSwan1964,
  author  = {Bass, Hyman and Heller, Alex and Swan, Richard},
  title   = {The Whitehead group of a polynomial extension},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume  = {22},
  pages   = {61--79},
  year    = {1964}
}

Prerequisites

03.02.22
03.08.01

Tier anchors

beginner: Milnor *Whitehead torsion* (Bull. AMS 72, 1966) §1 informal; Cohen *A Course in Simple-Homotopy Theory* Ch. I (collapses and expansions)
intermediate: Milnor *Whitehead torsion* §§3-5 (the torsion of an acyclic based complex; Wh(\pi)); Cohen Ch. II-IV; Rotman *Algebraic K-theory* on K_1
master: Milnor *Whitehead torsion* (Bull. AMS 72, 1966) full; Whitehead 1950 (*Ann. of Math.* 52); Kervaire 1965 (*Comment. Math. Helv.* 40, s-cobordism); Cohen *A Course in Simple-Homotopy Theory*; Lück *A Basic Introduction to Surgery Theory* §1

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Gradient-like vector fields, handle decompositions of a cobordism, and the self-indexing Morse function underlying the handle chain complex
Milnor — Whitehead torsion (Bulletin of the American Mathematical Society 72, 1966) · pp. 358--426; the Whitehead group Wh(\pi)=K_1(\mathbb{Z}\pi)/(\pm\pi), the torsion \tau of an acyclic based chain complex, simple homotopy equivalence, the s-cobordism theorem, and Wh(\mathbb{Z}/n)
Whitehead — Simple homotopy types (American Journal of Mathematics 72, 1950) · pp. 1--57; J.H.C. Whitehead's definition of simple homotopy type via elementary expansions and collapses, and the torsion obstruction in the group later named Wh(\pi)
Kervaire — Le théorème de Barden-Mazur-Stallings (Commentarii Mathematici Helvetici 40, 1965) · pp. 31--42; the s-cobordism theorem of Barden, Mazur, and Stallings: a compact h-cobordism of dimension >= 6 is a product iff its Whitehead torsion vanishes
Bass, Heller, Swan — The Whitehead group of a polynomial extension (Publ. Math. IHÉS 22, 1964) · pp. 61--79; the fundamental theorem of algebraic K-theory and the computation of Whitehead groups of free abelian groups and finite cyclic groups

Estimated time

beginner: 18m
intermediate: 46m
master: 84m