Detectors in homotopy theory
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1 Detectors in homotopy theory Mark Behrens University of Notre Dame
2 An analogy: Particle physics: Homotopy theory: All matter is built from elementary particles Topological spaces (up to homotopy) are built by attaching together disks (of varying dimensions) Goal: Discover all fundamental particles Tool: Massive accelerators and detectors [LHC] Goal: Compute all attaching maps (homotopy groups of spheres) Tool: Massive spectral sequences [Adams spectral sequence]
3 Matter: built out of elementary particles
4 CW complex: Built out of disks - D n n-cells
5 CW complexes Theorem: Every topological space is (weakly) homotopy equivalent to a CW complex. CW complexes have the form X = ڂ n X n X 0 = set of points X 1 = X 0 {set of intervals} X 2 = X 1 {set of disks} X i+1 = X i {set of D i+1 }
6 CW complexes Theorem: Every topological space is (weakly) homotopy equivalent to a CW complex. CW complexes have the form X = ڂ n X n X 0 = set of points X 1 = X 0 {set of intervals} X 2 = X 1 {set of disks} X i+1 = X i {set of D i+1 }
7 CW complexes Inductively, the CW complex X is determined up to homotopy by the homotopy classes of the attaching maps α π i (X i ) CW-complexes/homotopy = matter of geometry Building blocks elements of homotopy groups
8 Elementary particles: complicated [Wikipedia commons]
9 Homotopy groups of spheres: also complicated Computation: Serre, Toda, Chart: Hatcher
10 Down to business For the rest of this talk, all CW complexes are finite, connected, with fixed basepoint. We will discuss the simpler problem of classifying such CW complexes up to stable equivalence [still hard!]: X st Y Σ N X Σ N Y N 0 [define Σ] Stable homotopy category of these: Morphisms: X, Y st = Σ N X, Σ N Y N 0 (these stabilize) Stable equivalence of CW complex depends on stable attaching maps α π n st X S n, X st stable homotopy groups of X
11 Stable homotopy groups of spheres: π n st S n st π n+1 S n st π n+2 S n st π n+3 S n st π n+4 (S n )
12 Voltage bias Basic particle detectors: Takeaway: Use simple particle (electron) to detect more exotic particles [AMS-02 experiment]
13 Goal: build a detector of (stable) homotopy groups Homology a simple computable approximation of homotopy groups Hurewicz homomorphism: π st X H (X) [typically not an iso!] Homology classes will be the electrons in our detector which detect elements of π st (X)
14 Cellular homology Form a chain complex basis given by cells differential given by degrees of attaching maps:
15 Cellular homology Form a chain complex basis given by cells differential given by degrees of attaching maps:
16 Cellular homology Form a chain complex basis given by cells differential given by degrees of attaching maps: [ Graph notation]
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19 Cellular homology Form a chain complex basis given by cells differential given by degrees of attaching maps:
20 Cellular homology
21 Cellular homology
22 Cellular homology
23 Cellular homology: different coefficients Homology with coefficients in a ring R each dot represents a copy of R instead of a copy of Z. [In this example, R = F 2 ] BAD detector?? Does not detect degree 2 attaching maps!
24 Cohomology: reverse arrows
25 Cohomology: cup product structure Cohomology is a ring! In this example, H RP 4 ; F 2 = F 2 x /(x 5 ) Better detector: degree 2 attaching map detected by x 2 0 [If 2-cell attached to 1-cell with deg 0 map, would get x 2 = 0]
26 Cell diagrams: a way of encoding attaching maps
27 Cell diagrams: a way of encoding attaching maps Let j be minimal so that α factors through X j
28 Cell diagrams: a way of encoding attaching maps Suppose X has j-cells D 1 j, D 2 j, For each such cell D k j there is a projection map
29 Cell diagrams: a way of encoding attaching maps We say: the i-cell attaches to the j-cell D k j with attaching map α k
30 Cell diagrams: a way of encoding attaching maps We say: the i-cell attaches to the j-cell D k j with attaching map α k Note: - A given cell can attach nontrivially to many other cells - In this way, the stable equivalence class of X is essentially determined by the collection of all its attaching maps α k π st i 1 (S j ) - This is why I asserted that the stable homotopy groups of spheres are the elementary particles which comprise CW complexes
31 Cell diagrams: a way of encoding attaching maps Cell diagram: Refinement of Cellular chain complex 1) Draw one dot for each cell 2) Draw arrows labelled by attaching maps [examples: CP 2, RP 4 ]
32 Steenrod operations: more structure on mod 2 cohomology Theorem: (Steenrod) There are natural homomorphisms (i 0) Sq i : H n X; F 2 H n+i (X; F 2 ) x, Sq i?, x = x 2, 0 i = 0 1 i n 1 i = n i > n Steenrod operations sometimes detect attaching maps! [examples: CP 2, RP 4 ] Sq i Sq j = σ k j k 1 i 2k Sqi+j k Sq k [Adem relations] Steenrod algebra = algebra of these operators A = F 2 Sq i i > 0 / Adem relations Similar operations on H (X; F p )
33 For the rest of this talk, all cohomology reduced, with F 2 -coefficients! H X H (X; F 2 )
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37 Homology event channels Given: f: Y X (0) Direct detection Suppose the induced map is nonzero. f : H X H Y Define: signal f f Hom A H X, H Y = Ext A 0 (H X, H Y)
38 Homology event channels Given: f: Y X (1) Indirect detection single decay (zero on cohomology) Form a new CW complex - mapping cone C f X f CY [picture] The long exact sequence H X 0 H Y H +1 C f H +1 X 0 H +1 Y is actually a short exact sequence: 0 H Y H +1 C f H +1 X 0 If this extension of A-modules is nontrivial, get signal : 0 f Ext A 1 (H +1 X, H Y)
39 Homology event channels Given: f: Y X (s) Indirect detection s-decays Factor f into (zero on cohomology) Y = X 0 f1 X 1 f2 fs X s = X such that each f i is zero on cohomology - event Get an exact sequence 0 H Y H +1 C f1 H +2 C f2 H +s C fs H +s X 0 Get a signal f Ext A s (H +s X, H Y)
40 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Ext 0 cells not hit by Steenrod operations H RP 16 n [Chart: Ext computing software Bruner/Perry]
41 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Event: ι 1 : S 1 RP 16 Signal: F 2 n H RP 16
42 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Event: ι 3 : S 3 RP 16 Signal: F 2 n H RP 16
43 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Event: ι 7 : S 7 RP 16 Signal: F 2 n H RP 16
44 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Event: ι 1 η: S 2 RP 16 Signal: 0 F 2 H C ι1 η n H RP 16 0
45 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Event: S 3 η S 2 ι 1 η RP16 Signal: 0 F 2 H C η H C ι1 η n H RP 16 0
46 Examples of homology events, signals s f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Event: S 7 2 S 7 2 S 7 2ι 7 RP 16 Signal: 0 F 2 H C 2 H C 2 H C 2η n H RP 16 0
47 Two problems noise : f: Y X could be null homotopic, and yet produce a nonzero signal 0 f Ext A s H +s X, H Y physically impossible signals : For some signals x [f] for any f x Ext A s H +s X, H Y
48 s Adams differentials - Noise cancellation Turns out you can use physically impossible signals to cancel noise! f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Physically impossible event: there is no topological map S 15 ι 15 RP 16 Issue: 15-cell attaches to 7-cell with attaching map 2ι 7 σ: S 14 S 7 invisible to Steenrod operations! Signal: F 2 n H RP 16
49 s Adams differentials - Noise cancellation Turns out you can use physically impossible signals to cancel noise! f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) Noise: the event below gives a non-zero signal S 14 2 S 14 ι 7 σ RP16 Issue: the element 2σι 7 π st 14 (R 16 ) is null homotopicsince the 15-cell attaches to 7-cell with attaching map 0 2σ: S 14 S 7 F 2 2σι 7 extends over a disk! H C 2 Signal: H C ι7 σ n H RP 16 0
50 Adams differentials - Noise cancellation Turns out you can use physically impossible signals to cancel noise! f: S n RP 16 s The invisible attaching map 2σ: S 14 S 7 f Ext A s (H +s+n RP 16, F 2 ) simultaneously creates a physically impossible signal a noise signal. Noise cancellation cancel the noise with the correlated physically impossible signal. Adams differential Signal: n
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52 Adams differentials noise cancellation Turns out, there are differentials (Adams spectral sequence) d r : Ext A s H +s+n X, F 2 Ext A s+r H +s+r+n 1 X, F 2 d r impossible signal = noise Theorem (Adams) H Ext A H X, F 2, d r π st X 2 [2-completion = 2-torsion ] [Isomorphism of sets] (version for H ; F p p-completion)
53 s Adams differentials - Noise cancellation Turns out you can use physically impossible signals to cancel noise! f: S n RP 16 f Ext A s (H +s+n RP 16, F 2 ) n π st RP 16 Z : Z 2 Z 2 Z 8 Z 2 0 Z Z 2 Z Z 2 Z 8 Z Z 2 Z 2 Z Z 2 2 Z 2 Z 2 Z 8 Z Z 2 Z 2 Z 8 Z 2
54 s Adams differentials - Noise cancellation Only one little problem: you have no a priori knowledge what the differentials are! Only technique for deducing them: guile n π st RP 16 Z : Z 2 Z 2 Z 8 Z 2 0 Z Z 2 Z Z 2 Z 8 Z Z 2 Z 2 Z Z 2 2 Z 2 Z 2 Z 8 Z Z 2 Z 2 Z 8 Z 2
55 πst n+k (S k )
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57 Elementary particles of homotopy theory: [Wikipedia commons]
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60 Higher energies
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63 Higher energies require fancier detectors SLAC
64 Tevatron
65 LHC
66 ASS: mod 2 cohomology
67 Real K-theory ko [Lellmann-Mahowald] [Beaudry-B-Bhattacharya-Culver-Xu] Ext Abo (ko S k, ko S k+n )
68 Topological modular forms (tmf) [Mahowald] started thinking about the tmf-ass [B-Ormsby-Stojanoska-Stapleton] - A tmf [Beaudry-B-Bhattacharya-Culver-Xu] computing Ext Atmf [work in progress]
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