THE METHODS OF ALGEBRAIC TOPOLOGY FROM THE VIEWPOINT OF COBORDISM THEORY

May combine two claims. The quote only partly states this claim.

The representation a is faithful because it is already faithful on U*(P) subset U*(BU), where a coincides with [phi^{-1}(a)]*.

“The fidelity of the representation a follows from the fact that it is already faithful on U* (P)”

• Also stated in: § = α*, φ-.υ*(Βυ)^υ*(Μυ). · plain

  “C i/# (BU) by the preceding lemma, where a coincides with ιφ' 1 (a)] •”

May combine two claims.

Letting Lambda = O*(P) = Omega_0^O, there is a multiplication operator X -> alpha X for alpha in Lambda, and this defines a monomorphism Lambda -> A.

“This defines a monomorphism Λ -> A .”

May combine two claims.

The operations satisfy Delta_{(k1,k2)} d = 0, in particular d^2 = 0 and Delta d = 0; Delta Psi = 1; [d, chi] = 2; chi d = chi_1 d; and Theta Psi = 0.

“a) A (ki kl )°d = 0(in particular, d 2 = 0, Δ<9 = 0); b) ΔΨ= 1, [d, χ] = 2, X d = Xl od, where ^ = [CP 1 ] € Λ C A U ; ΘΨ = 0.”

May combine two claims.

Lemma 1.1: In the categories of direct and inverse spectra over S, short exact sequences exist that satisfy axiom 1 and the appropriate exactness axiom for Hom, and direct sums with amalgamation exist satisfying axiom 3.

“Lemma 1.1.In the categories S and S there exist short exact sequences 0-*/l->B->C^0, where A, B, C GS or A, B, C CS, satisfying axiom 1 of the category S and axiom 2 for the functor Horn* (T, ) if A, B, C C S and Τ 6 S, and axiom 2 for Horn* ( , T) if A, B, C € S and Τ £ S. In the categories S and S there exist direct sums with amalgamation satisfying axiom 3.”

May combine two claims.

The operation Phi associated with the Chern character has the properties Phi^2 = Phi, Phi*(1) = 1, Phi*(x) = 0 for dim x < 0, and Phi*: Lambda -> Lambda.

“1) Φ 2 = Φ, 2) Φ*(1)= 1, 3) Φ* (Λ;) = 0, dim» <0,Ϊ6Λ,Λ = U* (Ρ), Φ*: Λ -Λ, where2 5”

Classified as probably not a substantial claim.

The Steenrod ring A = GL(A) consists of infinite matrices a = (a_{omega,omega'}) indexed by nondyadic partitions, with entries in the ordinary Steenrod algebra A and with dimension given by lambda_omega - lambda_omega' + dim a_{omega,omega'}.

“A = GL (A) consists of infinite matrices a = (α ω ω ), where ω, ω' Site nondyadic partitions (a,, . . ., a s ), (a[, . . ., α^), α ω , ω' €• A and dim a = λ ω -λ ω '+ dim α ω< ω ' is the dimension of the matrix..”

May combine two claims.

The Adams spectral sequence with E_2 = Ext_{A_X}(X*(Y), X*(T)) exists when Y has an X-free acyclic filtration, or when Y and T are in S and X* is stable, Noetherian and acyclic.

“The so-called Adams spectral sequence (£ r , J r ) with £ 2 -term E 2 = Ext* Α χ(Χ*(Υ), Χ* (Τ)) arises in the following cases:”

May combine two claims.

If X = [CP^n], then sigma*_omega(X) = lambda_omega [CP^{n-k}] and sigma*(ab) = sum sigma_{omega1}(a) sigma_{omega2}(b) for a,b in Omega_U.

“If X = [CP n ], then σ* ω (Χ) = λ ω [CP n - k ] and σ*(αδ)= 2 ' σω,(α)σω 2 (6), a, b, e Q v .”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

In the category S⊗Z[1/2], all spectra kO coincide up to suspension, and the spectrum is a sum of two spectra of type k=kO+E^2 kO.

“It is easy to show that in the category S <8> χΖ [ l / 2 ] all spectra kO coincide up to suspension, and the spectrum It is a sum of two spectra of the type k = kO + E 2 kO .”

May combine two claims.

The subcategories D and D_p are not closed under the operations of axiom 1, but they are closed under those operations when the morphism f: A→B has f*: H*(B,Z)→H*(A,Z) an epimorphism.

“It should be noted that the subcategories D and D p ate not closed with respect to the operations entering in axiom 1 for S-categories.These subcategories, however, are closed with respect to the operations referred to, when the morphism /: A ^B is such that f*: H* (B, Z) -> H* (A, Z) is an epimorphism.”

May combine two claims.

O*(MSO) is a cyclic A-module with a single generator v in O^0(MSO), subject to the relations d(v) = 0 and Delta(v) = 0, and it has the displayed resolution.

“Further, it turns out that 0* (MSO) is a cyclic A -module with a single generator υ d 0* (MSO),”

• Also stated in: § A° = (A-S) · plain

  “given by the relations d(v) = 0, Δ (υ) = 0, and we have a resolution”

May combine two claims.

Criterion (A) applies in S_p when Y is a complex whose homotopy groups tensored with Q_p are finite, and in that case the Postnikov system gives a nonacyclic X-free resolution.

“*The criterion (A) for the convergence of the Adams spectral sequence applies easily in the category S p = S <S> zQp under the condition that Υ is a complex with ττ i * (Υ) ® Z Q P finite groups, in which case there is a nonacyclic resolution (the Postnikov system) which is X-ftee.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

Λ_p is an A^U-module and A^U is also a left Λ_p-module.

“Note that Λ ρ is an A ^-module and A u is also a left A p -module.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

In GL(A) there is a projection operator pi such that pi A pi = A, pi^2 = 1, and pi is in A_0 = GL(A).

“π such that π Α π = Α, π 2 = 1, π € A 0 = GL{A).”

May combine two claims.

The module U*(T) is finitely generated, and the theory U* is Noetherian.

“Therefore the module U* (T) is finitely generated and the theory U* is Noetherian.”

May combine two claims.

Lemma 3.2: For a stable spectrum K in D_p, X*(Y) is a free A-module, and the minimal dimension of its A-free generators equals n when Y is a spectrum of (n+m)-connected complexes.

“Lemma 3•2.Lei Κ € D p be a stable spectrum.Then X* (Y) is a free Α-module, where the minimal dimension of the Α-free generators is equal to n, if Υ = (Υ m ) is a spectrum of (n + m)-connected complexes Y m .”

May combine two claims.

If K=E^2 L, then Π^0(K)=X^1(K); hence in the S-category Π^0(K) is just X^1(K).

“If Κ = E 2 L, then Π 0 (K) = X 1 (K); therefore in the S-category Π 0 (K) is simply X 1 (K).”

May combine two claims. The quote only partly states this claim.

The operations Xi_omega in A have the properties listed in Lemma 5.6, including Xi_{(k)} a = a^{k+1} for a in Map(X, MU_1) and Xi_omega(a) = 0 for omega not equal to (k), and the composition of operations Xi_mu Xi_omega is a linear combination of operations Xi_omega with integral coefficients.

“Lemma 5.6.The operations Ξ ω £ A have the following properties:”

• Also stated in: § Hence · plain

  “1.If a € Map (X, MUJ CU 2 (X), then S (k) a = a k + 1 and S w (a)= 0 if ω φ (k). 4. The composition of operations S M -S a is a linear combination of operations of the form.S a with integral coefficients, so that an additive basis for the ring S consists of all δ ω .”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

There is a Thom isomorphism phi: U^i(X) -> U^{2n+i}(M_xi, *) for a complex n-bundle xi of dimension 2n, where M_xi is its Thom complex.

“Here there is also the Thom isomorphism φ: U l (X) -> U 2n + ' (Μξ, *) where ζ is a complex f rt -bundle of dimension 2n, and Μξ is its Thom complex.”

May combine two claims.

For primes p>2, the self-Ext ring of Λ_p equals the self-Ext ring of Z_p over the classical Steenrod algebra; the algebraic spectral sequence matches the geometric Adams sequence in mod p cohomology.

“Theorem.The ring Ext-J”

May combine two claims.

The ring Λ⊗Z Q_p is a local ring with maximal ideal m, and its quotient by m is Z_p.

“The ring over Q p , Λ ® Z Q p D Λ ρ , lies in A V <g) z Q p D A V, and Λ ® ζ Q p is a local ring with maximal ideal m C Λ ® z Q p , where h ® z Q p /m = Ζ p .”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

For any smooth quasicomplex manifold X, possibly with boundary, the Poincare-Atiyah duality law U*(X) = U_{n-i}(X, dX) holds.

“We recall that for any smooth quasicomplex manifold (possibly with boundary) there is the Poincare-Atiyah duality law”

• Also stated in: § A° = (A-S) · plain

  “U*(X) =<U n -i{X,dX)”

May combine two claims.

In an S-category L=E^2 L', Π^0(L) is P*(L); if L=EL', then Π^0(L) depends on the multiplication in P*(L').

“In particular, in an S-category L = E 2 L' we have that ΓΡ (L) is P* (L); in the case L = EL', Π 0 (L) depends on the multiplication in P* (L').”

May combine two claims.

Lemma 2.2: For X in S, the cohomology theories EX and X have the same Adams spectral sequences whenever those sequences exist; and if X is a direct sum of suspensions with indices tending to infinity, the resulting theory gives the same Adams spectral sequence as X.

“Lemma 2.2.a) If X G S, the cohomology theories EX and X have the same Adams spectral se--> <quences for any Υ and Τ for which the sequences exist (here Υ € S, Γ € S). b) Furthermore, if X = Σ Ε l Xis a direct sum, where γ. -» oo for i -> oo, fAere iAe theory X* defines the same Adams spectral sequence as the theory X*.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims. The quote only partly states this claim.

The Adams operations Psi^k in A_U tensor Z Z[1/p] satisfy the stated requirements, including Psi^k(xy) = Psi^k(x) Psi^k(y), Psi^k x = k^i x for x in Lambda^{-2i} = Omega_U^{-2i}, and Psi^k(u) = ...

“We define the Adams operations Ψ^ , which arise from the requirements:”

• Also stated in: § There arise classes X( klr k 2 ) · plain

  “i) Ψ(χν)Ψ 2) Ψυ • χ = tex • ψ£. r«e χ e Λ”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The theory P* generated by the point spectrum P=(S^0,*) acts on all cohomology theories, is called cohomotopy theory, has spheres as its spectrum, and is multiplicative because P⊗P=P.

“The previously mentioned theory P*, generated by the point spectrum Ρ = (S°, *), operates on all cohomology theories and is called”

The quote only partly states this claim.

For the (4n-2)-skeletons X_{2n} of the complexes MU_n, Ext^{1,j}(U*(X_{2n})) = 0 for t < 2n - dim T.

“From this and the lemma it follows that Ext l 'J (U* (A 2n )) = 0 for t < In-dim T,”

May combine two claims. The quote only partly states this claim.

Ext^{1,8k+6} = K Ext^{1,8k+2}, where K is in Ext^{0,?}(U*(MSU), Lambda); and d_3(Ext^{0,j}) = 0 for i not equal 8k+4, while d_3(Ext^{0,8k+4}) = Ext^{3,?} is defined by d_3(K) = h^3.

“Lemma 7.3. a) Ext 1 ', 8 * 46 = Κ Ext 1 •'** 2 .”

The quote only partly states this claim.

There exist operations a_{i,n} in Bψ such that a_{i,n}* = k^{n+i} for j<f(n), where f(n) tends to infinity as n grows.

“There exist operations a^j n G Β”

May combine two claims.

The subring Λ_{p,i}⊂Λ_p is invariant under the action of all relevant operations, and Λ_{p,i} is obtained from Λ_{p,i−1} in two steps: a central extension of k_{i+1} by Λ_{p,i−1}, then adjoining a polynomial generator y_i of dimension p^i−1 with specified action.

“The following fact is clear: the subring Λ ρ>; C Λ ρ is invariant with respect to the action of all Thus, the ring A Pii is obtained from the ring l p ,;-i in the following way (in two steps):”

May combine two claims.

There is a canonical isomorphism Ext_A(Z_p,Z_p)≅Ext_{A/(βA+Aβ)}(Z_p,Z_p) for t>1, where Z_p=H(P,Z_p), and the Hopf invariant q1(D) coincides with q1 for X1=K2=P.

“There is a canonical isomorphism ExtA {Ζρ,Ζρ) -ExtΑ/$Α+Α$(Ζρ,Ζρ) for t > 1, where Z p = Η (Ρ, Ζ ρ ), Ρ is a point, and the Hopf invariant q i (D) coincides with the Hopf invariant q l for X t = K 2 = P.”

May combine two claims.

The second spectral sequence is induced by the filtrations in Λ_p, A_p, and L and converges to Ext_?(L,Λ_).

“In the second, which is induced by the filtrations in Λ ρ , A p , L and which converges to Ext .υ (L, Λ_), we have:”

May combine two claims.

If a is an exact functor, then H^i(aC_X)=0 for i>0 and one may take C_Y=C_X and β^2=1; if in addition Ext_{A_B}( ,aN)∘e=0 for i>0, then aC_X=C_Y and Ext^1_Y(aM,aN)≅R^*(G_N(M)).

“In the following cases the group E l (a) coincides with the whole group Ext: a) α is an exact functor; here H'(aC x ) = 0, i > 0, and one can assume that Cy = Cy, /8 2 = 1.b) If in addition a is such that Ext^y ( , c7V) °e = 0 for i > 0, then aC^-= Cy and an isomorphism is generated:”

May combine two claims.

The formula P_k(v_p x)=v_p Φ s_ω Φ(x), with ω=(p−1,...,p−1) h times and P the Steenrod powers, is correct because all homomorphisms (Φ s_ω Φ)(y)=0 mod p when dim ω=dim y, i.e. (Φ s_ω Φ)^*(y)∈Ω_i^U=Z.

“We set”

May combine two claims. The quote only partly states this claim.

In the basic examples, the functors λ and v_p are exact in the category D; λ satisfies R^qG_N(M)=Ext^q_U(M,N) with M=U*(K2), N=U*(K1), and v_p satisfies R^qG_N(M)=Ext^q_{H*( ,Z_p)}(v_pM,v_pN).

“We have b) The functors λ and u p are exact in this category.”

Classified as probably not a substantial claim. May combine two claims.

Parts (a) and (b) follow from the form of the ring C, in particular from C⊗Z_p being commutative, from C being obtained from the standard resolution, and from H^*(C_i⊗Z_p)=Z_p[x]⊗Ext_{...}(Z_p,Z_p); the Bockstein β is derived from Lemma 11.3.

“The proof of (a) and (b) follows easily from the form of the ring C -in particular, from the fact that C ® Z p is commutative, C is obtained from the standard ©(-resolution and G>; <S> Z p has a system of generators [I ; i, while ff”

May combine two claims.

The image of the Hopf invariant q^{1,4k}:π_{n+4k−1}(S^n)→Ext^{1,4k}_{A_k}(k*(P),...) has index 2 for k=2l+1 and index 1 for k=2l; moreover, the image q^{1,k}(π_{n+4k−1}(S^n)) coincides with the image q^{1,k}·Jπ_{n+4k−1}(SO).

“Conclusion. The image of the Hopf invariant 1,4ft”

May combine two claims. The quote only partly states this claim.

For the groups Ext^{1,*}(Λ,Λ), the image of the Hopf invariant q1 is trivial for l=4k+3, an epimorphism for t=4k−1 and t=4k, and for l=4k+2 (k>1) and t=4k+3 (k≥0) the Adams differential d_3 is nontrivial, with d_3(E_2^{2,2t})=h_3E_2^{*,*} for t=4k+2 (k>1) and t=4k+3 (k>0).

“3) It was established which elements of all these groups Ext 1 are realized as images of the Hopf invariant q x .In particular, for the groups Ext 1 '”

May combine two claims.

If h is the nonzero element of Ext^{1,?}(U*(MSU), Lambda) = Z_2, then the homomorphism a -> ha from Ext^{i,?} to Ext^{i+1,?} is an epimorphism with kernel Im d* for i = 0 and an isomorphism for i > 1.

“c) If h £ Ext^'li (U* (MSU), Λ) = Z 2 is the nonzero element, then the homomorphism α -> ha: Ext ι ' υ -> Ext t+ .”

• Also stated in: § q-1 · plain

  “}' is an epimorphism with kernel Imd* for i = 0, and an isomorphism for i > 1 (we recall that the spectrum MSU is multiplicative).”

Classified as probably not a substantial claim. May combine two claims.

Each of the indicated spectral sequences defines a usual filtration i(x) on Hom(K,Y), and for the Adams spectral sequence of X this filtration is denoted i_X; there is a double filtration [i_X(x),φ(x)] with ψ(x)≤i_X(x).

“In addition, each of the indicated spectral sequences defines in the groups of homotopy classes of mappings Horn (K, Y) the usual filtration i(x), whose corresponding index i is such that the element χ £ Horn* {K, Y) is nontrivial in E 1 ^ and trivial in E'^ for / > i.For the Adams spectral sequence of the theory X we shall denote this filtration by ίχ.We have the double filtration ιίχ (x), φ(χ)],”

• AssumptionThe authors state this as an assumption: There is given a complex Y in S with a filtration whose associated A*-module complex is acyclic, has H_0=X*(Y), and whose modules are not assumed projective.

May combine two claims.

For all t>1, in H^{t,*}(C⊗Z_2) the kernel Ker β equals the image Im β; hence the reduction-modulo-p homomorphism a_p:Ext_C(Q_p[y],Q_p[y])→H^{*,*}(C⊗Z_p) is an isomorphism on Ker β=Im β, and none of these groups has elements of order p^2.

“e) For a/Z f > 1, in the groups H'' * (C; <S> Z 2 ) i/ie kernel Ker /3 coincides with the image Im Hence, the homomorphism of reduction modulo p, a P : Extc* «? p [y,L <? P [tfi])-»-^•• (C< ® Z p )”

May combine two claims. The quote only partly states this claim.

There is a composite map a=(...):E^1(a)→Ext^1_{A_B}(Y*(K1),Y*(K2)), where E^1(a) is a subset of Ext^1_{A_B}(X*(K2),X*(K1)) and E^1(a)=Γ^1∘β^1∘i(Ext^1_{A_B}(Y*(K1),Y*(K2))).

“We have the composite map”

May combine two claims.

From the group viewpoint λM=M⊗_ΛZ[x] for Λ-free M, so λ is exact and R^qλ=0 for q>0; for v_p, v_pU(P)=Z_p and v_pM=M⊗_ΛZ_p in D, and since U(K) and H(K) are free abelian, v_p is exact in D.

“From the group point of view we have”

May combine two claims. The quote only partly states this claim.

Part (c) follows from e_{i,h}(y_i)=... as shown in Lemma 11.1 and from the standard resolution F for Q_p/δ_i=Q_p; there is a natural homomorphism γ_p:Ext_{D,*}(Q_p[y],Q_p[y])→Ext_{U*}(U*(P),U*(P))⊗Z_Qp.

“Part (c) follows from the fact that e%,h{Vi ) == 2j { )p k x q ~h, as was shown in Lemma 11.1, and from the construction of the standard fe;-resolution F for the module ©;/δ; = Q p and the differential d* in the complex Horn* (Q p [yj • F, <? p tyj).”

May combine two claims.

In case (a), when a is exact, there is a spectral sequence (E_r,d_r) with E_2^{p,q}=R^pG^q_N(M) converging to Ext^{p+q}(aM,aN), and G^q_N(M)=Ext^q_{A_B}( ,eN)∘c; from this spectral sequence the homomorphism is a monomorphism.

“In case (a) (a is an exact functor) there arises a spectral sequence (E r , d T ), where Εζ' q -2 R p G q , N (M), which converges to Ext** (aM, a/V), and G Qi N (M) = Ext?. * ( , e”

May combine two claims.

If X^i(P)=0 for i>0 and X^0(P)=π, then there is an augmentation functor ν:X*→H*( ,π), and for any group G a functor ν_G:X*→H*( ,π⊗G).

“2. If the theory X* is such that X 1 (P) = 0 for i > 0 and Χ”

May combine two claims.

The image of the complexification homomorphism c in Ext^{1,*}(Λ,Λ) has index 2 for l=8k+4, index 1 for l=8k, and is zero otherwise.

“Since in the groups k c (P) the image of the homomorphism c has index 2 for t = 8k + 4, index 1 for t = 8k and is equal to zero for t^ 8k, 8k + 4,we can draw from this the conclusion that the image group Im'c C Ext 1 '^ (Λ, Λ) has index 2 for I = 8k + 4, index 1 for I = 8k and is equal to zero for ι Φ 8k, Α ψ, 8k + 4, since Im^ = Im'c'.”

May combine two claims.

As a Bψ-module, k*(P) has one generator t and is given by the relations (p^n Ψ_p)(t)=p^n t for all primes p and large n.

“The module k*(P), as a βψ.-module, has a single generator f and is given by the relations (ρ”

May combine two claims.

The groups H*(W, d*) satisfy H^{8k}(W) = H^{8k+4}(W) = Z_2 + ... + Z_2, with the number of summands equal to the number of partitions of k, and H^i(W) = 0 for i not equal 8k or 8k+4.

“H ik {W) = H ski4 (W) = Z 2 +”

• Also stated in: § q-1 · plain

  “. ..+Z 2 (the number of summands is equal to the number of partitions of the integer k), Η ^W) = 0, i φ 8k, 8k + 4.”

May combine two claims.

If a in E_3^{p,q} for p > 3 and d_3(a) = 0, then a = d_3(beta). Hence E_4^{p,q} = 0 for p > 3, and V = F.

“Lemma 7.2.// a. € Εξ• q for ρ > 3, and d 3 (a) = 0, then a = d i (β).Hence £/• q = 0 for ρ > 3, and V = F (see [13])”

• cites [13] 13. M. Kervaire and J. Milnor, Bernoulli numbers, homotopy groups and a theorem of Rohlin, Proc. Internal. Congr. Math., 1958, Cambridge Univ. Press, New York, (I960), pp. 454-458. MR 22 #5537.Not checked yet

  Not checked in the free run.

May combine two claims.

In the category D of complexes without torsion, the operator chi^n: k^{2n}(X) -> k^0(X) has image exactly the elements of K^0(X) whose filtration is at least 2n, and chi is a monomorphism.

“In the category D of complexes without torsion the operator x n :k 2n (X) -> k°(X) is such that its image consists precisely of all elements in h°(X) = K°(X) whose filtration is > In; moreover, χ is a monomorphism.”

May combine two claims. The quote only partly states this claim.

Ext^{1,?}(U*(MSU), Lambda) = Ext^{1,8k+6}(U*(MSU), Lambda), and Ext^{i,?}(U*(MSU), Lambda) = 0 for i not equal 8k+2 or 8k+6.

“ΕκΙ^2 (U' (MSU), A) = Exti' h+6 (IP (MSU), Λ)”

• Also stated in: § q-1 · plain

  “and (U*(MSU),A) = 0, i¥=8k + 2, 8k + 6.”

May combine two claims.

There is an element h in Ext^{0,1}_{Bψ}(A0,A0) with 2h=0 and h^3=0, and multiplication by h is a monomorphism for s=8k and s=8k+1.

“There is defined an element h € Ext 0 ' 1 n (AQ, AQ) such that 2h = 0, h 3 = 0, while multiplication”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The category of A-modules corresponding to D is the category of Λ-free modules, where Λ=U*(P)≈Sly.

“The category of A -modules corresponding to the category D is the category of Λ-free modules, where Λ = U* (P) » Sly.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims. The quote only partly states this claim.

The ring Bψ^{kO} is built from operators constructed from Ψ and multiplication operators on A0=kO*(P), with commutativity relations Ψ^k h = k h Ψ^k, Ψ^k v = k^2 v Ψ^k, and Ψ^k w = k^4 w Ψ^k.

“The ring Β ψ is composed, in a fashion identical to that for the ring Β ψ , from such operators a = (a n ) constructed from Ψ and from the multiplication operators on A o = kO*(P), keeping in mind the following commutativity relations:”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

If B consists of a single object K, then A_B=End X*(K) and there is no Hopf invariant.

“If Β consists of a single object K, then .4g= YLndX* {K) and there is no Hopf invariant.”

May combine two claims.

The ring Q_p[y_i]·B_{p,i} coincides with Ί_{p,i}, and the commutation rules for y_i and Φ_i s_ω Φ_i follow from part (b) of Lemma 11.1.

“The ring Q p [y t”

May combine two claims.

There exist operations δ_i in Aψ of dimension 2i, with δ_0=1, whose χ^i δ_i form an additive topological basis of Aψ, and every element of Aψ is a formal integer series in these basis elements.

“It is easy to see that it is possible to construct operations δ; G Αψ of dimension 2t, where δ 0 = 1, such that the elements χ δ ; give an additive topological basis for the ring Αψ , and all elements of Αψ can be described as formal series SA^x^g^ where the Aj. are integers.”

May combine two claims.

On the cohomology of a point, λ is such that Λ=Z[x] and λ(y)=T(y)x^i, where y∈Ω_i^U=U^{2i}(P) and T is the Todd genus.

“On the cohomology of a point Λ the functor λ is such that Λ-Ζ [xl and”

May combine two claims.

In the category B∩B' the operation of the ring BL is well-defined as an extension Bψ→βψ, because torsion in A0=kO*(P) prevents the operation from being determined by its representation on kO*(P), unlike k*-theory in D.

“In the category Β D B' the operation of the ring BL is well-defined, the latter being α priori an extension Βψ -» β ψ , since in view of the presence of torsion in AQ -kO*{P) the operation is not defined by its own representation on kO*(P), in contrast to &*-theory in the category D.”

May combine two claims. The quote only partly states this claim.

For t=1 the homomorphism γ_p is a monomorphism; for all p>2 the homomorphism γ_p^{1,*} is an isomorphism; for p=2 it is an isomorphism on Ext^{1,2q} for q=2 and q odd, while for q=2s, s>2, its image has index 1 or 2 in Ext^{1,2q}⊗Z_Qp, and in fact index 2 for all q=4s, s>1.

“Theorem 11.2.a) For t = 1 the homomorphism γ ρ is a monomorphism.”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The cohomology ring of a point in kO-theory, kO*(P)=A0, has generators 1,h,v,w and relations h^2=0, h^3=0, hv=0, v^2=4w.

“The cohomology ring of a point kO (P) = AQ is described as follows: generators 1”

The quote only partly states this claim.

The groups Ext^{1,2i}(Λ,Λ) are isomorphic to Z_2 for i=4l+1.

“Thus, we have proved Theorem 10.1.The groups Ext'' 2 j(A, Λ) are isomorphic to Z 2 for i = ll + 1.”

May combine two claims.

The geometric dimension of the homotopy groups equals q-m-Is, which equals qs-t.

“is equal to q -m -Is = qs -t.”

May combine two claims.

For the point spectrum X=P, the Aψ-module k*(P) has one generator t and is given by the relations δ_i(t)=0 for i>0.

“Example 1.Let X = Ρ 6 Ό be the point spectrum.Then k* (P) has a single generator t as an ^ψmodule and is given by the relations δ ; (ί) = 0, i > 0.”

May combine two claims.

The polynomial generators of Λ_p=U_p(P)=ΦU(P) can be chosen as rational-coefficient polynomials in the elements x_i=[CP^{p^i}]∈Ω_U, and in the first nontrivial dimension p−1 the generator can be identified with [CP^{p−1}]_x.

“We note that the polynomial generators of the ring Λ ρ = U p (Ρ) = Φ U (P) can be chosen to be polynomials with rational coefficients in the elements x i = [CP P”

May combine two claims.

In the category B the spectral sequence for kO* is trivial, and there is a subcategory B' in which the operation of the ring Ω^ψ is well-defined.

“1) The spectral sequence (E n d r ) j kO* is trivial in B; in Β there is a subcategory β”

The quote only partly states this claim.

In k*⊗Q, the same operations Ψ are obtained as formal sums Σ μ_i χ^i δ_i = Ψ, with μ_i rational and δ_i integral for large n and i<f(n).

“Remark 8.1.The same operations Ψ in k* ® Q are obtained as formal sums of the form Σμ ί χ ί δ £ = Ψ”

Probably not a substantial claim: may not pass S4 (based).

The groups Ext^1_{Bψ}(k(MU),k*(P)) are nontrivial even for t=1.

“Remark 8. 2. In what follows it will become clear that the groups Ext 1 '! (k (MU), k*(P))arenon-Α”

Probably not a substantial claim: may not pass S4 (based).

The ring Λ_{p,i} generated by the first i polynomial generators y_1,...,y_i in Λ_p does not depend on the choice of generators.

“This ring Λ ρ>ΐ does not depend on the choice of generators.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In the second example with objects K1,K2 and L=CK1∨K2, the ring A_B consists of endomorphisms of X(L) preserving the image β*X*(K1)⊂X(L), and the Hopf invariant q1_B(g) is defined and vanishes exactly when X*(L)_B=X*(K1)+X*(K2) as groups.

“2. If Β consists of objects K lt K 2 , L = CK 1”

Borderline on S4 (based): unclear from the text. May combine two claims.

There is a natural homomorphism r^q:Ext^q_{A_B}(M,N)→R^qG_N(M), with r=Σr^q, and homomorphisms β^q:R^qG_N(M)→H^q(Hom_{Bψ}(C_Y,aN)), where Ker β^2=0.

“There is defined a”

May combine two claims.

For the functor λ, the action of the operators (k^nΨ) and multiplication by x in k-theory are calculated by A and λ, proving part (a) of the lemma for λ.

“Thus, the action of the operators (&”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing), S4 (based).

There is a complexification operator c:kO*→k* with c(h)=0, c(v)=2x^2, c(w)=x^4, where x is the Bott periodicity operator.

“We have the”

May combine two claims.

For large n, the integer γ_i used to define the operation does not depend on n, so the operation is well-defined.

“It is easy to see that the number γ ι does not depend on re for large re -> oo.Hence the operation is welldefined.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

For X=U, the augmentation functors v and v_p and the Riemann-Roch functor λ preserve the ring structure of the theory.

“For the theory X = U , the augmentation functors v, v p and the Riemann-Roch functor λ preserve the ring structure of the theory.”

May combine two claims.

In A⊗Z_Qp there exists a projector Φ such that Φ(x,y)=Φ(x)Φ(y), and Im Φ*⊂Λ is the polynomial ring in generators y_i of dimension 2p^i−2, where the y_i are polynomial generators of Λ=Ω_U⊗Z_Qp with σ_ω(y) divisible by p and σ_ω(x_i)=p, k=p^i−1.

“Earlier, in §5, it was proved that in the algebra A 1 <S> zQp there exists a projector Φ €.A <£)zQp such that Φ(χ, γ) = Φ(%)Φ(ν) and ΙπιΦ* C Λ is the ring of polynomials in generators y,, . . ., y _ _ _ ; dim γι = 2p”

May combine two claims.

The image of the reduction-modulo-p homomorphism a_p is generated by A_j x^{q−1} for p>2 and all q, by h_1 x^{q−1} for q=2 and q=1 mod 2, and by A_1 x^{q−1}+A_2 x^{q−2} for p=2 and q=0 mod 2.

“d) The image of the homomorphism of”

May combine two claims.

For all s > 1, there are isomorphisms Ext^s(U*(MSU), Lambda) = H_{t-2s}(W, d'), and Ext^{1,t}(U*(MSU), Lambda) = Ker d* intersect Ker Delta* in Omega_U.

“Lemma 7.1.a) For all s > 1, we have isomorphisms”

• Also stated in: § q-1 · plain

  “X (£/'(MSU),A) = Ht-2 s(W, d'). b) ExtlV (IT (MSU), A) = Ker d* Π Ker Δ* c Ώ ν .”

May combine two claims. The quote only partly states this claim.

In the cobordism example, the filtration Y⊃Y_0⊃... is acyclic though not free in X=H( ,Z_p); the filtration i(x) is a homotopy invariant equal to i_{U_p}(x), and Ext_Q(Z_p,Z_p)=A_p=Z_p[h_0,h_1,...,h_i,...].

“The most important example which we consider here is the theory of cobordism in the category the filtration Υ D Y o 3 ... is also acyclic (although not free) in the theory X = Η ( , Z p ).”

• AssumptionThe authors state this as an assumption: There is given a complex Y in S with a filtration whose associated A*-module complex is acyclic, has H_0=X*(Y), and whose modules are not assumed projective.

The quote only partly states this claim.

The torsion part of Ext^1_{Bψ}(k*(P),k*(P)) is cyclic, with order equal to the greatest common divisor of the sequence |p^n(p^l−1)|_p for large n and prime p.

“Lemma 8.3.The torsion part of the group Ext 1 '£'(k* (P), k* (P)) is a cyclic group, whose order is equal to ip”

May combine two claims.

There is a Bott operator chi: k^{2n} -> k^{2n-2} defined by Bott periodicity, with Omega^2 k_{2n} = k_{2n-2}, and k_{2n} is a connective fiber of BU.

“2 k 2n = k 2n . 2 ,”

• Also stated in: § *^8. /c-theory in the category of complexes without torsion · plain

  “and k 2n is a connective fiber of BU.”

Classified as probably not a substantial claim. May combine two claims.

The Atiyah-Hirzebruch-Grothendieck element λ^(n) in K^0(MU_n) defines maps λ_1:U*→K* and λ:U*→k*, where λ=(λ^(n)) is the unique element in k^{2n}(MU_n) such that x^n λ^(n)=1 in K^0(MU_n), with x the Bott operator.

“3. The Riemann-Roch functor.Let X = U and 7 = A: ; we consider the Atiyah-Hirzebruch-”

May combine two claims.

The collection of projectors pi_i = sum lambda_i A_{y_i} - y_{i+1} satisfies pi_i pi_k = 0 for i not equal k and decomposes U* tensor Q into a sum of identical theories.

“Moreover, the collection of projectors π £ = J l A l y -y l +1 is such that πρ^ = 0, / Φ k, and it decomposes the theory U* ® Q into a sum of identical theories.”

The quote only partly states this claim.

The basis element u_{1,8k+4} of Ext^{1,8k+4} satisfies d_3(u_k)=h^3 Ext^{1,8k+2}(Λ,Λ), with h in Ext^{0,1}(Λ,Λ)=Z_2.

“The basis element u^. of the group Ext 1<8 y +4 is such that i,8fe 1,2”

The quote only partly states this claim.

In kO*-theory one can introduce operations (k^n Ψ^k) and their combinations, and the resulting ring of operators is identical to the analogous ring in k*-theory lying in Aψ.

“In the theory kO* it is possible by analogy with the theory k* to introduce operations (k nX”

May combine two claims. The quote only partly states this claim.

For u,v in V(X), the product u@v has the form u+v+2Σ x_{i,j} u^i v^j, with coefficients independent of u and v; the product is commutative, associative, and has an inverse.

“the law of multiplication u@v = f(u, v) has the form”

May combine two claims. The quote only partly states this claim.

For X=K(Z)+EK(Z), the ordinary Adams spectral sequence is zero, while for Y=K(Z_q), d_q(v)=h_0 u.

“for X the ordinary Adams spectral sequence is zero, but for Υ we have: d q (v ) = h^u , where u C E xt °/ and ν € Ext/, since 77* (Υ) = Ζ q .”

May combine two claims.

The groups Ext^{1,*}(Λ,Λ)=Z_2 are cycles for all Adams differentials and belong to the image of the Hopf invariant q_U.

“The groups Ext 1 '^.(Λ, Λ) = Z 2 are cycles for all Adams differentials and belong to the image of the Hopf invariant qiu”

May combine two claims.

The image of the homomorphism α_p from Ext_J^* coincides with Ker β in all dimensions greater than 1, while Ker α_p is nontrivial in dimension 1.

“The image of the homomorphism a p (ExtJ*) coincides with Ker β in all dimensions > 1, although Ker a p is nontrivial in dimension 1 Lsee parts (c) and (d)J.”

May combine two claims.

Lemma 11.2(c) states that Ext^{1,t}(Q_p[y], Q_p[y]) is nontrivial for t=2p(p^i-2), i>1, and is isomorphic to Ext_{Z_p[x]⊗Λ[h_i]}, the cyclic group Z_p.

“c) The group Ext i ^t(Qp LyJ, Q p LyJ) is nontrivial for t = 2p(p' -2), </ > 1, ara</ is isomorphic to Ext Z p [x]®A[h u the cyclic group Zf |”

May combine two claims.

In Lemma 11.1(a), the subring f_i commutes with all multiplication operators from A_{p,i-1} in Λ_p, and these are all operators Φ_i Φ S_ω Φ Φ_i for ω=(p^{i1}-1,...,p^{is}-1) with j_k < i-1.

“a) The subring fej commutes with all operators of multiplication A. piimi C Λ ρ are</ a// operators Φ^Φδα,ΦΦ; /or οΖί ω = (ρ ; Ί -1, . . ., p /s -1), z^Aere j k <i -I.”

Classified as probably not a substantial claim.

The Adams operations Ψ^_q are defined by ψ^_q(a)=q^{-m} a^{ψ^q} for dim a=2m.

“ψ£ (α) = q-m aW£, dim a = 2m”

Probably not a substantial claim: may not pass S3 (load-bearing).

The classical Adams spectral sequence with second term E_2=Ext^{s,t}_A(H*(X),Z_p) for X in S_gr has differentials arranged as d:F^{s,h}->F^{s+r,h+r-1}.

“The classical Adams spectral sequence with second term E 2 ' = Ext4' '^(//* (A), Z p ) for X Ε S gr is arranged as follows:”

May combine two claims.

For K=P, the representation a -> [φ^{-1}(a)]* of the Steenrod ring A_U in endomorphisms of U*(P)=Ω_U is dual by Poincare-Atiyah duality to the operation of A on U*(P) and is faithful.

“is dual by Poincare-Atiyah duality to the operation of the ring A on U* (P) and is a faithful representation.”

May combine two claims.

The representation a -> a of the ring A on U*(BU) is well-defined and faithful.

“Lemma 5.3• ΤΆβ representation a -> ο of the ring A on U* (BU) is well-defined and is faithful.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The element h_0 in Ext^{0,1}(Z_p,Z_p) is associated with multiplication by p.

“h 0 Ε Ext^0 1 (Z p , Z p ) is associated with multiplication by ρ”

May combine two claims.

The spectral sequence (E_r,d_r) is cruder than the Adams spectral sequence in H*( ,Z_p)-theory, and E_∞ is larger than stable homotopy groups of spheres; therefore the cobordism Adams spectral sequence can in principle be nontrivial because E_∞ is associated with Ext_{Λ^U}(Λ,Λ)⊗Z Q_p.

“A priori the spectral sequence (E n a r ) is cruder than the Adams spectral sequence in H*{ , Z p )-theory and £*o** is bigger than the stable homotopy groups of spheres; on account of this, the Adams spectral sequence for cobordism theory constructed in this work can in principle be non-trivial, since Coo is associated with Ext /ll/ (A j Λ) <g> z Q p -”

May combine two claims.

The E_2 term of the algebraic Adams spectral sequence is canonically isomorphic to Ext_A(Z_p,Z_p), which is the second term of the usual topological Adams spectral sequence.

“From this it follows that the algebra E 2 for the”

May combine two claims.

In the study of Ext_{A_U}(U*(MSU)), all dimensions are doubled, the groups E_i^{s,k+1} for 1 < i < 3 are constructed identically, the element h_0 in Ext^{1,1} is replaced by h in Ext^{1,?}, and the Adams differential d_3 is nontrivial.

“In the study of Ext AU (U* (MSU)) all dimensions will be doubled, the groups E^s k+l for 1 < i < 3 will be constructed in an identical fashion, but the element h 0 € Ext 1 ' 1 will be replaced by an element h € Ext 1 '^, and the Adams differential d 3 will be non-trivial (see § §6, 7).”

May combine two claims.

Working as in §§9 and 12, these theories give the best filtrations for homotopy groups, at least for complexes without torsion; for p=2 the appropriate substitute for MU may be MSU.

“Besides, working as in § §9 and 12, it is possible to convince oneself that these give the best filtrations for homotopy groups (at any rate, for complexes without torsion; for ρ = 2 it may be that the appropriate substitute for MU is MSU).”

May combine two claims.

It is possible to prove nonexistence of elements of Hopf invariant one by extraordinary cohomology methods and also to calculate Adams differentials.

“In this way it is possible not only to prove the nonexistence of elements of Hopf invariant one by the methods of extraordinary cohomology theory as in [4] (see also § §9, 10), but also to calculate Adams differentials.”

• cites [4] 4. L4J J. F. Adams and M. F. Atiyah, K-theory and the Hopf invariant, Quart. J. Math. Oxford Ser. (2) 17 (1966), 31-38. MR 33 #6618. sourceNot checked — no full text of the source

May combine two claims.

Consequently, D is closed under the construction of X-free acyclic resolutions (only acyclic), and the Adams spectral sequence can be studied for X,Y in D or D_p.

“Therefore the category D is closed under the construction of X-itee acyclic resolutions (only acyclic), and it is possible to study the Adams spectral sequence only for Χ, Υ G D (or D p ).”

May combine two claims.

In the Steenrod algebra for p>2, besides the usual grading there is a second grading counting Bockstein occurrences, and this grading has no analogue for p=2.

“As is shown in H. Cartan's well-known work, the Steenrod algebra for ρ > 2 in addition to the usual grading possesses a second grading (”

• cites H. Cartan's well-known workNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

For p>2, the cohomology Ext_A(Z_p,Z_p) carries a triple grading, whereas for p=2 it does not.

“Therefore for ρ > 2 the cohomology Ext/i (Z p , Z p ) has a triple grading in distinction to ρ = 2.”

May combine two claims.

The algebraic Adams spectral sequence associated with the topological one, beginning with E_2=Ext(Z_p,Z_p) and converging to Ext(Λ,Λ)⊗ZQ_p, is algebraically well-defined and non-trivial a priori.

“is algebraically well-defined, is non-trivial a priori.”

Probably not a substantial claim: may not pass S4 (based). The quote only partly states this claim.

If the analogous bigradation conjecture holds for other functors, then the entire classical Adams spectral sequence and stable homotopy groups of spheres for p>2 can be calculated by unitary cobordism theory.

“the entire classical Adams spectral sequence and the stable homotopy groups of spheres for ρ > 2 can be completely calculated by means of the theory of unitary cobordism”

Probably not a substantial claim: may not pass S1 (asserted), S3 (load-bearing), S4 (based). May combine two claims. The quote only partly states this claim.

The equation π_*^{(p)}(S)≈Ext(Λ,Λ)⊗ZQ_p for p>2 would imply triviality of the author's Adams spectral sequence in U-cobordism and absence of extensions in E∞=E_2.

“the triviality of the Adams spectral sequence constructed by the author in the theory of i7-cobordism; b) the absence of extensions in the term Eco = E 2 .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Theorems 1 and 2 can be derived from the Adams spectral sequence in cobordism theory and therefore do not depend on the preceding conjectures.

“The author has available a derivation of Theorems 1 and 2 from the Adams spectral sequence in cobordism theory, and hence Theorems 1 and 2 do not depend on the preceding conjectures.”

May combine two claims.

For p>2, Ext^{s,t}_A(Λ,Λ)⊗ZQ_p consists of cycles for all differentials, and elements of Ext^{1,*} are realized by elements of π_i^{(p)}(S^N) of the same order.

“Ext''^(A, h)®zQp consists (for ρ > 2) of cycles for all differentials, while elements of Ext 1 '^ are realized by elements of πί ρ) (S N ) of the same order”

Probably not a substantial claim: may not pass S4 (based).

If the conjecture is true, then the spectra of points and complexes without p-torsion in homology belong to the category S_gr.

“If the conjecture is true, then the spectra of points (spheres) Ρ and complexes without p-torsion in homology belong to the category S gr .”