AN INTRODUCTION TO EXOTIC SPHERES AND SINGULARITIES Andrew Ranicki - PowerPoint PPT Presentation

1 AN INTRODUCTION TO EXOTIC SPHERES AND SINGULARITIES Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ � aar Edinburgh, 4 May, 2012

2 The original papers ◮ J. Milnor, On manifolds homeomorphic to the 7-sphere , Annals of Maths. 64, 399-405 (1956) ◮ M. Kervaire and J. Milnor, Groups of homotopy spheres I. , Annals of Maths. 77, 504–537 (1963) ◮ F. Pham, Formules de Picard-Lefschetz g´ en´ eralis´ ees et egrales , Bull. Soc. Math. France 93, ramification des int´ 333-367 (1965) ◮ E. Brieskorn, Beispiele zur Differentialtopologie von Singularit¨ aten , Inventiones math. 2, 1–14 (1966) ◮ F. Hirzebruch, Singularities and exotic spheres , Seminaire Bourbaki 314, 1966/67 ◮ J. Milnor, Singular points of complex hypersurfaces , Annals of Maths. Study 61 (1968)

3 Homotopy spheres ◮ A homotopy m -sphere Σ m is a differentiable oriented m -dimensional manifold which is homotopy equivalent to S m . ◮ For m � 5 Σ m is homeomorphic to S m . ◮ Σ m is standard if it is diffeomorphic to S m . ◮ Σ m is exotic if it is not diffeomorphic to S m . ◮ In this lecture will describe the construction and main properties of the Brieskorn spheres , which arise as the links of the isolated singularities of complex hypersurfaces.

4 The original exotic spheres ◮ The original exotic 7-spheres Σ 7 of Milnor (1956) were constructed as boundaries Σ 7 = ∂ F of the ( D 4 , S 3 )-bundles over S 4 ( D 4 , S 3 ) → ( F , ∂ F ) → S 4 of the 4-plane vector bundles over S 4 classified by particular elements in π 4 ( BSO (4)) = Z ⊕ Z . ◮ The exotic nature of Σ 7 detected by the defect signature( F ) − ⟨L ( F ) , [ F ] ⟩ ∈ Q of the Hirzebruch signature theorem for an 8-dimensional manifold F with ∂ F = Σ 7 . ◮ Kervaire and Milnor (1963) showed that there are 28 differentiable structures on S 7 .

5 Bounding exotic spheres ◮ A homotopy m -sphere Σ m bounds if Σ m = ∂ F for a framed ( m + 1)-dimensional manifold F . ◮ Pairs ( F , ∂ F ), ( F ′ , ∂ F ′ ) are cobordant if there exists an = ∂ F ′ such that orientation-preserving diffeomorphism ∂ F ∼ F ∪ ∂ − F ′ is a framed boundary. The cobordism classes constitute a group bP m +1 under connected sum. ◮ Kervaire-Milnor (1963) computed bP m +1 to be a quotient of the simply-connected surgery obstruction group P m +1 = L m +1 ( Z ) . ◮ No obstruction to simply-connected odd-dimensional surgery, P 2 n − 1 = L 2 n − 1 ( Z ) = 0, so that bP 2 n − 1 = 0: every bounding homotopy (2 n − 2)-sphere Σ 2 n − 2 is standard.

6 The bounding odd-dimensional homotopy spheres I. ◮ Every bounding homotopy (2 n − 3)-sphere is the boundary Σ 2 n − 3 = ∂ F of an ( n − 2)-connected framed (2 n − 2)-dimensional manifold F 2 n − 2 constructed by plumbing together µ copies of τ S n − 1 using a nonsingular ( − 1) n − 1 -quadratic form ( H n − 1 ( F ) = Z µ , b , q ) over Z . ◮ The rel ∂ surgery obstruction of ( F , ∂ F ) → ( D 2 n − 2 , S 2 n − 3 ) is { signature( F ) / 8 σ ( F ) = Kervaire( F ) { Z if n is odd ∈ P 2 n − 2 = L 2 n − 2 ( Z ) = if n is even . Z 2 ◮ Kervaire( F ) = Arf( H n − 1 ( F ; Z 2 ) , q ) is the Arf invariant of the quadratic form q determined by the framing. ◮ The surjection b : P 2 n − 2 → bP 2 n − 2 ; σ ( F ) �→ ∂ F is a precursor of the Wall realization of surgery obstructions. ◮ The groups bP 2 n − 2 are cyclic finite.

7 The bounding odd-dimensional homotopy spheres II. ◮ bP 4 m is cyclic of order σ m / 8 with σ m = ϵ m 2 2 m − 2 (2 2 m − 1 − 1)numerator( B m / 4 m ) where B m is the m th Bernoulli number, and ϵ m = 2 or 1, according as to whether m is odd or even. ◮ bP 8 = Z 28 , generated by one of the Milnor 1956 examples. ◮   0 if there exists a framed  = bP 4 m +2 (4 m + 2)-dimensional manifold   otherwise Z 2 { 0 for m = 0 , 1 , 3 , 7 , 15 = for m ̸ = 0 , 1 , 3 , 7 , 15 , 31 . Z 2 ◮ bP 126 = Z 2 or 0.

8 The Brieskorn-Hirzebruch-Pham-Milnor construction ◮ For any a = ( a 1 , a 2 , . . . , a n ) with a 1 , a 2 , . . . , a n � 2 the map P a : C n → C ; ( z 1 , z 2 , . . . , z n ) �→ z a 1 1 + z a 2 2 + · · · + z a n n has an isolated singularity at a (0) = complex hypersurface ⊂ C n . (0 , 0 , . . . , 0) ∈ P − 1 ◮ The ‘(star,link)’-pair of the singularity is a framed (2 n − 2)-dimensional manifold with boundary ( F , ∂ F ) ⊂ C n constructed near the singular point. The complexity of the singularity is measured by the differential topology of ( F , ∂ F ). ◮ A Brieskorn sphere is a link ∂ F = Σ 2 n − 3 which happens to be a homotopy (2 n − 3)-sphere, necessarily bounding. ◮ Σ 2 n − 3 can be exotic.

9 The hypersurface Ξ a ( t ) ◮ Terminology of Brieskorn (1966) ◮ For t ∈ C define the hypersurface = P − 1 Ξ a ( t ) a ( t ) = { ( z 1 , z 2 , . . . , z n ) ∈ C n | z a 1 1 + z a 1 1 + · · · + z a n n = t } ⊂ C n ◮ Ξ a ( t ) is non-compact if n � 2. ◮ For t ̸ = 0 Ξ a ( t ) is nonsingular, an open (2 n − 2)-dimensional manifold, with a diffeomorphism Ξ a ( t ) ∼ = Ξ a (1) . ◮ Write Ξ a (1) = Ξ a .

10 The star F a and link Σ a of the singular point (0 , 0 , . . . , 0) ∈ Ξ a (0) ◮ Ξ a (0) has an isolated singularity at (0 , 0 , . . . , 0), with Ξ a (0) \{ (0 , 0 , . . . , 0) } an open (2 n − 2)-dimensional manifold ◮ For t ̸ = 0 the star of the singularity is the compact framed (2 n − 2)-dimensional manifold F a ( t ) = Ξ a ( t ) ∩ D 2 n ⊂ D 2 n . ( F a ( t ) denoted M a ( t ) by Brieskorn). ◮ The link of the singularity is Σ a ( t ) = ∂ F a ( t ) = Ξ a ( t ) ∩ S 2 n − 1 ⊂ S 2 n − 1 ◮ For t ̸ = 0 with | t | sufficiently small the (star, link) pair is independent of t , and written ( F a ( t ) , Σ a ( t )) = ( F a , Σ a ) , with a diffeomorphism F a \ ∂ F a ∼ = Ξ a .

11 The Milnor fibration ◮ The codimension 2 submanifold ( F a , Σ a ) ⊂ ( D 2 n , S 2 n − 1 ) is framed, i.e. extends to an embedding ( F a , Σ a ) × D 2 ⊂ ( D 2 n , S 2 n − 1 ) . ◮ Define the (2 n − 1)-dimensional manifold with boundary ( E a , ∂ E a ) = (cl.( S 2 n − 1 \ Σ a × D 2 ) , Σ a × S 1 ) . ◮ The Milnor fibration map z a 1 1 + z a 2 2 + · · · + z a n p : E a → S 1 ; ( z 1 , z 2 , . . . , z n ) �→ n ∥ z a 1 1 + z a 2 2 + · · · + z a n n ∥ is the projection of a fibre bundle with fibre p − 1 (1) = F a . ◮ The monodromy automorphism h : F a → F a is such that E a = F a × I / { ( x , 0) ∼ ( h ( x ) , 1) | x ∈ F a } with p : E a → S 1 ; [ x , θ ] �→ e 2 π i θ and h | = id. : ∂ F a = Σ a → Σ a , p | = proj. : ∂ E a = Σ a × S 1 → S 1 .

12 The join ◮ The join of topological spaces A , B is the space A ∗ B = ( A × I × B ) / { ( a 1 , 0 , b ) ∼ ( a 2 , 0 , b ) , ( a , 0 , b 1 ) ∼ ( a , 0 , b 2 ) } for all a , a 1 , a 2 ∈ A , b , b 1 , b 2 ∈ B . ◮ If the reduced homology groups ˜ H ∗ ( A ), ˜ H ∗ ( B ) are without torsion then ∑ ˜ H i ( A ) ⊗ ˜ ˜ H r +1 ( A ∗ B ) = H j ( B ) . i + j = r ◮ If A is non-empty, and B is path-connected, then A ∗ B is simply-connected. ◮ The join is associative, with a homeomorphism ( A ∗ B ) ∗ C ∼ = A ∗ ( B ∗ C ) .

13 The algebraic and differential topology of ( F a , Σ a ) I. ◮ Pham, Brieskorn, Hirzebruch and Milnor determined the algebraic and differential topology of ( F a , Σ a ), in particular the conditions under which Σ a is a homotopy sphere, and determined the differentiable structure. ◮ The subspace of Ξ a = { ( z 1 , . . . , z n ) ∈ Ξ a | z a j Ξ real is real for j = 1 , 2 , . . . , n } a j has the following properties. ◮ Ξ real is a compact deformation retract of Ξ a = F a \ Σ a . a ◮ Ξ real = G 1 ∗ G 2 ∗ · · · ∗ G n is the join of the cyclic groups a G j = Z a j of order a j , regarded as discrete spaces with a j elements. ◮ Ξ real is ( n − 2)-connected, with homotopy equivalences a ≃ Ξ a ≃ F a ≃ S n − 1 ∨ S n − 1 ∨ ... ∨ S n − 1 Ξ real a involving µ = ( a 1 − 1)( a 2 − 1) . . . ( a n − 1) copies of S n − 1 . µ is called the Milnor number , with H n − 1 ( F a ) = Z µ .

14 The algebraic and differential topology of ( F a , Σ a ) II. ◮ The characteristic polynomial of the monodromy automorphism h ∗ : H n − 1 ( F a ) → H n − 1 ( F a ) is ∆ a ( z ) = det( z − h ∗ : H n − 1 ( F a )[ z ] → H n − 1 ( F a )[ z ]) ∏ n ∏ ( z − ω i 1 1 ω i 2 2 . . . ω i n = n ) ∈ Z [ z ] k =1 0 < i k < a k with ω j = e 2 π i / a j ∈ S 1 . ◮ For n � 4 Σ a is ( n − 3)-connected, with exact sequence 0 → H n − 1 (Σ a ) → H n − 1 ( F a ) 1 − h ∗ � H n − 1 ( F a ) → H n − 2 (Σ a ) → 0 . Thus Σ a is a homotopy (2 n − 3)-sphere if and only if ∆ a (1) = 1 ∈ Z .

15 The Kervaire invariants of Brieskorn (4 m + 1) -spheres ◮ J. Levine, Polynomial invariants of codimension two , Annals of Maths. 84, 537–554 (1966) ◮ For m � 1 let a = ( a 1 , a 2 , . . . , a 2 m +2 ) be such that Σ a is a homotopy (4 m + 1)-sphere. The Kervaire invariant of F a in L 4 m +2 ( Z ) = { 0 , 1 } is σ ( F a ) = Arf( H 2 m +1 ( F a ; Z 2 ) , q ) { 0 if ∆ a ( − 1) ≡ ± 1 mod 8 = 1 if ∆ a ( − 1) ≡ ± 3 mod 8