Spectra and G spectra

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1 September 23, ecp/latex/talks/intro-to-spectra.pdf

2 Cell structures Definition A cell structure on a pointed space X is an inductive presentation by iteratively attaching n disks: S n 1 X (n 1) D n X (n).

3 Cell structures Definition A cell structure on a pointed space X is an inductive presentation by iteratively attaching n disks: S n 1 X (n 1) D n X (n). Suspension Σ is an operation on spaces which preserves gluing squares, and ΣS n 1 S n and ΣD n D n+1. So, Σ is a shift operator on cell structures.

4 Cell structures Definition A cell structure on a pointed space X is an inductive presentation by iteratively attaching n disks: S n 1 X (n 1) D n X (n). Suspension Σ is an operation on spaces which preserves gluing squares, and ΣS n 1 S n and ΣD n D n+1. So, Σ is a shift operator on cell structures. Theorem ( Stability ) H n (X ; A) = H n+1 (ΣX ; A), ΣH (X ; A) = H (ΣX ; A).

5 Suspension: Freudenthal s theorem Calculation: π of a suspension n π n S 1 Z π n+1 ΣS 1 Z Z Z/2 Z/2 Z/12 Z/2 Z/2 Z/3...

6 Suspension: Freudenthal s theorem Calculation: π of a suspension n π n S 1 Z π n+1 ΣS 1 Z Z Z/2 Z/2 Z/12 Z/2 Z/2 Z/3... Theorem (Freudenthal) X : s connected space (π s X = 0) Y : t connective space (π <t Y = 0) Then F (Y, X ) F (ΣY, ΣX ) is a (2s t) equivalence.

7 Suspension: Freudenthal s theorem Calculation: π of a suspension n π n S 1 Z π n+1 ΣS 1 Z Z Z/2 Z/2 Z/12 Z/2 Z/2 Z/3... Theorem (Freudenthal) X : s connected space (π s X = 0) Y : t connective space (π <t Y = 0) Then F (Y, X ) F (ΣY, ΣX ) is a (2s t) equivalence. Corollary The 2 matters: π n F (Σ m Y, Σ m X ) is independent of m n.

8 Suspension spectra Definition Call Σ X the suspension spectrum of X. [Σ Y, Σ X ] = colim m [Σ m Y, Σ m X ]

9 Suspension spectra Definition Call Σ X the suspension spectrum of X. [Σ Y, Σ X ] = colim m [Σ m Y, Σ m X ] = colim m [Y, Ω m Σ m X ] = [Y, colim m Ω m Σ m X ] =: [Y, QX ].

10 Suspension spectra Definition Call Σ X the suspension spectrum of X. [Σ Y, Σ X ] = colim m [Σ m Y, Σ m X ] = colim m [Y, Ω m Σ m X ] = [Y, colim m Ω m Σ m X ] =: [Y, QX ]. Σ SuspensionSpectra Ω Spaces Q Spaces.

11 The Eilenberg Mac Lane spectrum Good news: stable invariants π Σ X = [Σ S, Σ X ] is a stable invariant of X.

12 The Eilenberg Mac Lane spectrum Good news: stable invariants π Σ X = [Σ S, Σ X ] is a stable invariant of X. On the other side, the sequence QΣ X represents a stable functor. This is because QΣX deloops QX : Ω(QΣX ) = QX. Hence, [ΣY, QΣ X ] = [Y, ΩQΣ X ] = [Y, QΣ 1 X.]

13 The Eilenberg Mac Lane spectrum Good news: stable invariants π Σ X = [Σ S, Σ X ] is a stable invariant of X. On the other side, the sequence QΣ X represents a stable functor. This is because QΣX deloops QX : Ω(QΣX ) = QX. Hence, [ΣY, QΣ X ] = [Y, ΩQΣ X ] = [Y, QΣ 1 X.] Bad news: not all stable invariants K(A, n) represents a stable functor too: [Y, K(A, n)] = H n (Y ; A). K(A, n + 1) deloops K(A, n), but K(A, n) QX for any X.

14 The Eilenberg Mac Lane spectrum A if = n, π Σ K(A, n) = 0 if 2n, n, mystery groups if > 2n. So, colim n Σ n Σ K(A, n) has the right homotopy groups.

15 The Eilenberg Mac Lane spectrum A if = n, π Σ K(A, n) = 0 if 2n, n, mystery groups if > 2n. So, colim n Σ n Σ K(A, n) has the right homotopy groups. Definition (Boardman, more or less) A spectrum is an ind-diagram of things like Σ n Σ X. The Eilenberg Mac Lane spectrum is presented by the ind-system HA := {Σ n Σ K(A, n)}.

16 Smash product, representability Theorem (Boardman) The smash product lifts from spaces to spectra: {Σ nα Σ X α } {Σ m β Σ Y β } =: {Σ nα+m β Σ (X α Y β )}. It has an adjoint, the function spectrum: [Z Y, X ] [Z, X Y ].

17 Smash product, representability Theorem (Boardman) The smash product lifts from spaces to spectra: {Σ nα Σ X α } {Σ m β Σ Y β } =: {Σ nα+m β Σ (X α Y β )}. It has an adjoint, the function spectrum: [Z Y, X ] [Z, X Y ]. Theorem X π (HA Σ X ) and X π (HA Σ X ) satisfy the axioms of ordinary (co)homology with A coefficients.

18 Smash product, representability Theorem (Boardman) The smash product lifts from spaces to spectra: {Σ nα Σ X α } {Σ m β Σ Y β } =: {Σ nα+m β Σ (X α Y β )}. It has an adjoint, the function spectrum: [Z Y, X ] [Z, X Y ]. Theorem X π (HA Σ X ) and X π (HA Σ X ) satisfy the axioms of ordinary (co)homology with A coefficients. Theorem (Brown, Atiyah) For E ( ) and E ( ) generalized (co)homology theories, there is a spectrum E such that Ẽ (X ) = π (E Σ X ) and Ẽ (X ) = π (E Σ X ).

19 Gluing homology theories Moral Spectra are an enrichment of homology theories where homotopy theory can be done.

20 Gluing homology theories Moral Spectra are an enrichment of homology theories where homotopy theory can be done. Example: Quotient sequences The quotient sequence S 2 S S/2 induces an exact sequence 0 π 2 S π 2 S/2 π 1 S 0 0 Z/2 Z/4 Z/2 0. Spectra guarantee that these problems have consistent solutions.

21 Ring spectra Ring-valued cohomology theories induce multiplication maps on representing spectra. In homotopy theory, associativity is a structure rather than a property: (a (b c)) ((a b) c)

22 Ring spectra Ring-valued cohomology theories induce multiplication maps on representing spectra. In homotopy theory, associativity is a structure rather than a property: (a (b c)) ((a b) c) S 0 F (E 3, E)

23 Ring spectra Ring-valued cohomology theories induce multiplication maps on representing spectra. In homotopy theory, associativity is a structure rather than a property: (a (b c)) ((a b) c) S 0 F (E 3, E) ((a b) c) d (a b) (c d) (a (b c)) d a ((b c) d) a (b (c d))

24 Ring spectra Ring-valued cohomology theories induce multiplication maps on representing spectra. In homotopy theory, associativity is a structure rather than a property: (a (b c)) ((a b) c) S 0 F (E 3, E) ((a b) c) d (a b) (c d) (a (b c)) d S 1 F (E 4, E). a ((b c) d) a (b (c d))

25 Ring spectra Ring-valued cohomology theories induce multiplication maps on representing spectra. In homotopy theory, associativity is a structure rather than a property: (a (b c)) ((a b) c) S 0 F (E 3, E) ((a b) c) d (a b) (c d) (a (b c)) d S 1 F (E 4, E). a ((b c) d) a (b (c d))..

26 Ring spectra Ring-valued cohomology theories induce multiplication maps on representing spectra. In homotopy theory, associativity is a structure rather than a property: (a (b c)) ((a b) c) S 0 F (E 3, E) ((a b) c) d (a b) (c d) (a (b c)) d S 1 F (E 4, E). a ((b c) d) a (b (c d)).. Leads to quasicategories and A rings ( coherently associative ). It pays off: A rings have a good theory of modules,....

27 Generalized cellular chains Theorem (Atiyah Hirzebruch) Let E be a generalized homology theory and X a cellular space. E 1 p,q = C cell p (X ; E q ) E p+q X.

28 Generalized cellular chains Theorem (Atiyah Hirzebruch) Let E be a generalized homology theory and X a cellular space. E 1 p,q = C cell p (X ; E q ) E p+q X. A cell structure suspends to a presentation of Σ X by shifts of wedges of S. Applying E to these diagrams give a presentation of E Σ X by shifts of wedges of E.

29 Generalized cellular chains Theorem (Atiyah Hirzebruch) Let E be a generalized homology theory and X a cellular space. E 1 p,q = C cell p (X ; E q ) E p+q X. A cell structure suspends to a presentation of Σ X by shifts of wedges of S. Applying E to these diagrams give a presentation of E Σ X by shifts of wedges of E. For E = HA, there is a sense in which HA Σ X C (X ; A). E Σ X E chains on X. In good cases, this is base change from S to E.

30 Intermission

31 Basics of equivariant homotopy theory Where spaces had points, G spaces have orbits: G/H equivariant X. Different choices of H G stratify the space: G/H F G (G/H +, X ) = X H.

32 Basics of equivariant homotopy theory Where spaces had points, G spaces have orbits: G/H equivariant X. Different choices of H G stratify the space: G/H F G (G/H +, X ) = X H. Definitions π n (X ) : G/H [G/H + S n, X ] G = π n X H A weak equivalence of G spaces is a G map which is a π iso. That is, for each H π X H π Y H.

33 Equivariant obstruction theory Definition A G cell structure on a pointed G space X is a presentation by iteratively attaching n disks of the form G/H + D n along images of G/H + S n 1.

34 Equivariant obstruction theory Definition A G cell structure on a pointed G space X is a presentation by iteratively attaching n disks of the form G/H + D n along images of G/H + S n 1. We would like a cohomology theory that controls the obstructions to extending maps of G cell complexes across a new cell, analogous to the role of ordinary cohomology.

35 Equivariant obstruction theory Definition A G cell structure on a pointed G space X is a presentation by iteratively attaching n disks of the form G/H + D n along images of G/H + S n 1. We would like a cohomology theory that controls the obstructions to extending maps of G cell complexes across a new cell, analogous to the role of ordinary cohomology. C n (X ; M) : G/H Hom(H n ((X H ) n, (X H ) n 1 ), M(G/H)). Satisfies the obvious Eilenberg Steenrod axioms.

36 Equivariant obstruction theory Definition A G cell structure on a pointed G space X is a presentation by iteratively attaching n disks of the form G/H + D n along images of G/H + S n 1. We would like a cohomology theory that controls the obstructions to extending maps of G cell complexes across a new cell, analogous to the role of ordinary cohomology. C n (X ; M) : G/H Hom(H n ((X H ) n, (X H ) n 1 ), M(G/H)). Satisfies the obvious Eilenberg Steenrod axioms. Warning This works, but it s not great. No Poincaré duality, for instance.

37 Bredon cohomology Question Sphere could also mean S V := V + for V a G representation. Spheres grade cohomology theories: S n H n. When can a representation be put in for in H (X ; M)?

38 Bredon cohomology Question Sphere could also mean S V := V + for V a G representation. Spheres grade cohomology theories: S n H n. When can a representation be put in for in H (X ; M)? Answer Exactly when M is a Mackey functor: for any G map we choose a transfer map f : G/H G/K t(f ) : M(G/H) M(G/K) satisfying a double coset formula reminiscent of character theory. (The definition is set up so that G/H Rep(H) fits.)

39 Bredon cohomology Question Sphere could also mean S V := V + for V a G representation. Spheres grade cohomology theories: S n H n. When can a representation be put in for in H (X ; M)? Answer Exactly when M is a Mackey functor: for any G map we choose a transfer map f : G/H G/K t(f ) : M(G/H) M(G/K) satisfying a double coset formula reminiscent of character theory. (The definition is set up so that G/H Rep(H) fits.) These are great: Poincaré duality and everything else you could hope for.

40 G spectra Definitions, redux Define suspension G spectra by [Σ G Y, Σ G X ] G = [Y, colim V Ω V Σ V X ] G. Equivariant Freudenthal says this colimit is degenerate. G spectra are ind-systems of S V desuspensions of suspension G spectra.

41 G spectra Definitions, redux Define suspension G spectra by [Σ G Y, Σ G X ] G = [Y, colim V Ω V Σ V X ] G. Equivariant Freudenthal says this colimit is degenerate. G spectra are ind-systems of S V desuspensions of suspension G spectra. Theorem, redux For E ( ) and E ( ) generalized Bredon (co)homology theories (i.e., = V is allowed), there is a G spectrum E such that Ẽ (X ) = π (E Σ G X ) and Ẽ (X ) = π (E Σ G X ).

42 G spectra Definitions, redux Define suspension G spectra by [Σ G Y, Σ G X ] G = [Y, colim V Ω V Σ V X ] G. Equivariant Freudenthal says this colimit is degenerate. G spectra are ind-systems of S V desuspensions of suspension G spectra. Theorem, redux For E ( ) and E ( ) generalized Bredon (co)homology theories (i.e., = V is allowed), there is a G spectrum E such that Ẽ (X ) = π (E Σ G X ) and Ẽ (X ) = π (E Σ G X ). Theorem, redux For any Mackey functor M, there is an Eilenberg Mac Lane G spectrum HM presenting Bredon cohomology H ( ; M).

43 Stable fixed points We built G spaces so that they carry fixed point data: X H. This splits into three notions of fixed points for G spectra: Geometric: Φ H (Σ G X ) = Σ X H, Φ H (colim α {X α }) = colim α {Φ H X α }, Φ H (X Y ) = Φ H (X ) Φ H (Y ). Categorical: [E, X H ] = [E, X ] H, π n (X ) : G/H π n X H. Homotopical: X hh = F H (EH +, X ).

44 Stable fixed points We built G spaces so that they carry fixed point data: X H. This splits into three notions of fixed points for G spectra: Geometric: Φ H (Σ G X ) = Σ X H, Φ H (colim α {X α }) = colim α {Φ H X α }, Φ H (X Y ) = Φ H (X ) Φ H (Y ). Categorical: [E, X H ] = [E, X ] H, π n (X ) : G/H π n X H. Homotopical: X hh = F H (EH +, X ). There is a map of fiber sequences? X H Φ H (X ) X hh X hh X th. Generally, this is the best we can say.

45 Stable fixed points We built G spaces so that they carry fixed point data: X H. This splits into three notions of fixed points for G spectra: Geometric: Φ H (Σ G X ) = Σ X H, Φ H (colim α {X α }) = colim α {Φ H X α }, Φ H (X Y ) = Φ H (X ) Φ H (Y ). Categorical: [E, X H ] = [E, X ] H, π n (X ) : G/H π n X H. Homotopical: X hh = F H (EH +, X ). There is a map of fiber sequences X hh if H = C p? X H Φ H (X ) transfer homotopy orbits X hh X hh X th. Tate spectrum Generally, this is the best we can say.

46 Advertisement: KO KU hc 2 KU exists as a C 2 spectrum with action by complex conjugation. X hh X H Φ H (X ) KO KO X =KU = H=C 2 X hh X hh X th. KO KO.

47 Advertisement: KO KU hc 2 KU exists as a C 2 spectrum with action by complex conjugation. X hh X H Φ H (X ) KO KO X =KU = H=C 2 X hh X hh X th. KO KO. Homotopy fixed point spectral sequence: H gp(c 2 ; π KU) π KO η2η3... η [β 2 ] [β 2 ]

48 Advertisement: KO KU C 2 Slice spectral sequence (Dugger) You can also get the homotopy groups as Mackey functors.

49 Advertisement: THH Theorem (McCarthy) Let f : R S be a surjection of rings with nilpotent kernel. Then there is a pullback square K(R) p trace TC(R) p where ( TC(R) = fib trace K(S) p TC(S) p, lim n THH(R)C p n R id lim n THH(R)C p n ) and THH is the subject of this (and the Thursday) seminar.

50 Advertisement: THH Theorem (McCarthy) Let f : R S be a surjection of rings with nilpotent kernel. Then there is a pullback square K(R) p trace TC(R) p where ( TC(R) = fib trace K(S) p TC(S) p, lim n THH(R)C p n R id lim n THH(R)C p n ) and THH is the subject of this (and the Thursday) seminar. There are lots of theorems along these lines, relating equivariant structure on THH to sundry things in algebraic K theory.

The Ordinary RO(C 2 )-graded Cohomology of a Point

The Ordinary RO(C 2 )-graded Cohomology of a Point Tiago uerreiro May 27, 2015 Abstract This paper consists of an extended abstract of the Master Thesis of the author. Here, we outline the most important