Constructive Homological Algebra VI. Constructive Spectral Sequences

Transcription

1 0 Constructive Homological Algebra VI. Constructive Spectral Sequences Francis Sergeraert, Institut Fourier, Grenoble, France Genova Summer School, 2006

2 1 [F F τ B B] Key tools for effective spectral sequences in topology: Eilenberg-Zilber theorem. C (F B)??? C (F ) C (B) Twisted Eilenberg-Zilber theorem. C (F τ B)??? C (F )??? C (B)

3 2 Eilenberg-Zilber theorem. (almost) canonical reduction: EZ F,B = (f, g, h) : C (F B) C (F ) C (B) f = Alexander-Whitney map (linear complexity). g = Eilenberg-MacLane map = Decomposition p q in simplices (exponential complexity). h = Shih Weishu map (exponential complexity).

4 3 Particular case F = B = 7. Eilenberg-Zilber reduction C ( 7 7 ) C N ( 7 ) C N ( 7 ) n , ,872 1, ,524 4, ,944 9,527 n 5 759,752 11, ,549,936 12, ,360,501 11, ,703,512 8, ,322,180 4,368 n 10 1,475,208 1, , , , ,432 1

5 4 Comparison between C (F B) and C (F τ B)? Same simplices Same underlying graded modules. Different incidence relations Differential perturbation. Basic perturbation lemma can be applied. Theorem (Edgar Brown + Shih Weishu): (f, g, h) : C (F τ B) C (F ) t C (B) for t = some algebraic tensor product twist.

6 5 Example 1. Effective homology version of the Serre spectral sequence. F = (F, C (F ), EC F, εf ) + B = (B, C (B), EC B, εb ) + τ : B F Serre EH E = F τ B = (E, C (E), EC E, ε E ) (Serre + G. Hirsch + H. Cartan + Shih W. + Szczarba + Ronnie Brown + J. Rubio + FS)

7 Proof. 6 C (F B) id C (F B) EZ C F C B C F C B ĈF ĈB EC F EC B Serre EH C (F τ B) id C (F τ B) Shih C F t C B C F t C B EP L ĈF t Ĉ B BP L EC F t EC B + Composition of equivalences = O.K.

8 7 Serre s canonical loop space fibration. I := [0, 1] P (X, ) := C([I, 0]; [X, ]) =: P X Ω(X, ) := C([I, 0, 1]; [X,, ]) =: ΩX Canonical fibration : ΩX P X X Combinatorial Kan version Genuine principal fibration: GX [E P X = GX τ X] X

9 Similar algebraic fibration. 8 [C = coalgebra] + [M + N = C-comodules] Cobar C (M, N). Particular case: Cobar C (C, Z) Z. Algebraic fibration: Cobar C (Z, Z) Cobar C (C, Z) Cobar C (Z, Z) t C C = Algebraic translation of: GX GX τ X X (simplicial) ΩX P X X (topological)

10 9 Example 2: Julio Rubio s solution of Adams problem. X = (X, C (X), EC X, εx ) Eil.-Moore EH ΩX = (ΩX, C (ΩX), EC ΩX, ε ΩX ) = Trivial iteration now available.

11 Proof (Step 0): 10 Three algebraic versions for the path space Serre fibration: ΩX P X X. C (GX) C (GX) Cobar C (X) (Z, Z) C (GX τ X) C (GX) t C (X) Cobar C (X) (C (X), Z) C (X) C (X) C (X) Simplicial Mixed Algebraic where GX = Kan model for the loop space ΩX.

12 11 Proof (Step 1): C (GX X) EZ C (GX) C (X) BPL 1 1 C (GX τ X) Shih C (GX) t C (X) GX τ X contractible 2 C (GX τ X) Z C (GX) t C (X) H Z

13 12 Proof (Step 2): [C (GX) ] BPL 2 Cobar C (X) (C (X), Z) Z Cobar C (X) (C (GX) C (X), Z) C (GX) Cobar C (X) (C (GX) t C (X), Z) C }{{} (GX) Key object

14 13 Proof (Step 3): Step 1 C (GX) t C (X) H Z Cobar C (X) (C (GX) t C (X), Z) Cobar C (X) (Z, Z) for the trivial C (X)-comodule structure, but BPL 3 Cobar C (X) (C (GX) t C (X), Z) Cobar C (X) (Z, Z) for the canonical C (X)-comodule structure.

15 14 Proof (Step 4): Step 2 Cobar C (X) (C (GX) t C (X), Z) C (GX) Step-3 Cobar C (X) (C (GX) t C (X), Z) Cobar C (X) (Z, Z) Step-2 + Step-3 C (GX) Cobar C (X) (Z, Z)

16 15 Proof (Step-5): C (X) ĈX EC X Cobar C (X) (Z, Z) CobarĈX (Z, Z) Cobar ECX (Z, Z) for the trivial coalgebra structures Cobar C (X) (Z, Z) EP L CobarĈX (Z, Z) BP L 4 Cobar ECX (Z, Z) for the right (A -) coalgebra structure Cobar C (X) (Z, Z) Cobar ECX (Z, Z).

17 16 Proof (Step-6): Step-4 Step-5 C (GX) Cobar C (X) (Z, Z) Cobar C (X) (Z, Z) Cobar ECX (Z, Z) Step-4 + Step-5 + Composition of equivalences C (GX) Cobar ECX (Z, Z) Q.E.D.

18 Very simple solution of Adam s problem : 17 Indefinite iteration of the Cobar construction??? X = (X, C (X), EC X, εx ) Ω EH ΩX = (ΩX, C (ΩX), EC ΩX, ε ΩX ) Ω EH Ω 2 X = (Ω 2 X, C (Ω 2 X), EC Ω2 X, ε Ω2X ) Ω EH Ω 3 X = (Ω 3 X, C (Ω 3 X), EC Ω3 X, ε Ω3X ) Ω EH Ω 4 X =... Cobar 3 (EC X)

19 Example of CA-Spectral Sequence. 18 Computation of the homotopy groups of S 2 P R = = Infinite real projective space stunted at dimension 2 := P R/P 1 R. S 0 S 1 S 2 S 3 S P 0 R P 1 R P 2 R P 3 R P R Elementary: H 2 = Z π 2 = Z consider the fibration: K(Z, 1) [X 3 := K(Z, 1) τ S 2 P R] S 2 P R H (X 3 )=???

20 Beginning of the Serre spectral sequence. 19 H (K(Z, 1)) = (Z, Z, 0, 0,...(1-periodic)) H (S 2 P R) = (Z, 0, Z, Z 2, 0, Z 2, 0,... (2-periodic)) E 2, (page 2) : Z 0 Z Z 2 0 Z 2 0 d 2 = Z 0 Z d 2 = Z 2 0 Z 2 d 2 = 0 H 3 =??? Note the non-trivial extension problem: 0 Z??? Z 2 0 automatically solved by Kenzo.

21 Simplex diagram for the generator s3? of H 3 (X 3 ). 20 s31 0 s s s32 2 s22 s3? = two 3-cells glued along their boundary = 3-sphere. QED.

22 21 s31 s21 s32 s11 s22

23 Gunnar Carlsson + James Milgram, in Handbook of Algebraic Topology, 1995: 22 Stable Homotopy and Iterated Loop Spaces In Section 5 we showed that for a connected CW complex with no one cells one may produce a CW complex, with cell complex given as the free monoid on generating cells, each one in one dimension less than the corresponding cell of X, which is homotopy equivalent to ΩX. To go further one should study similar models for double loop spaces, and more generally for iterated loop spaces.... /...

24 23... /... In principle this is direct. Assume X has no i-cells for 1 i n then we can iterate the Adams-Hilton construction of Section 5 and obtain a cell complex which represents Ω n X. However, the question of determining the boundaries of the cells is very difficult as we already saw with Adam s solution of the problem in the special case that X is a simplicial complex with sk 1 (X) collapsed to a point. It is possible to extend Adams analysis to Ω 2 X, but as we will see there will be severe difficulties with extending it to higher loop spaces except in the case where X = Σ n Y.

25 0 The END Francis Sergeraert, Institut Fourier, Grenoble, France Genova Summer School, 2006