Proof. The degree of the differentials follow from the construction of the exact couple for the spectral sequence. We also have the map $X_+\to S^{0}$ which gives a map $E \to F^{S^{1}}_{tr}(X_+,\,E)$ which is $S^{1}$-equivariant. Thus we have a map between the homotopy fixed point spectral sequences which maps the classes $y$ to one another, and so, (1) follows from the identification in example 2.5.

It remains to prove (3). For this, we show that for $k$ sufficiently large, $y^{k}$ lies in the image of a differential. It will then follow that for $r$ sufficiently large the classes $y^{m}$ and their $\pi _\ast E$ multiples are $0$ in the $E_r$-page for $m\geq k$. Therefore, the $E_r$-page will be concentrated in the columns between $1$ and $k$, and $E_\infty =E_r$ by increasing $r$ further if necessary. Hence, the spectral sequence converges strongly [Reference Boardman6, theorem 7.4].

The space $X/S^{1}$ being finite dimensional implies that the classifying map $X/S^{1} \to BS^{1}$ (for the $S^{1}$-bundle $X\to X/S^{1}$) factors through a finite skeleton. Hence, we have an equivariant map $X\to S({{\mathbb {C}}}^{k+1})$ for some $k$, and thus a map $F^{S^{1}}_{tr}(S({{\mathbb {C}}}^{k+1})_+,\,E) \to F^{S^{1}}_{tr}(X_+,\,E)$. As $E$ is complex orientable,

\[ \pi_\ast F^{S^{1}}_{tr}(S({{\mathbb{C}}}^{k+1})_+,E) \cong \pi_\ast F({{\mathbb{C}}}P^{k}, E) \cong E^{-{\ast}}({{\mathbb{C}}}P^{k}) \cong \pi_\ast E [y]/(y^{k+1}) \]

for some choice of complex orientation $y$. Observe that the homotopy fixed point spectral sequence for the space $ES^{1}$ as in example 2.5 matches with the Atiyah-Hirzebruch spectral sequence for ${{\mathbb {C}}}P^{\infty }$. It follows that the class $y$ represents the complex orientation in the $E_2$-page. For $S({{\mathbb {C}}}^{k+1})$ and hence also for $X$ via the equivariant map $X \to S({{\mathbb {C}}}^{k+1})$, the class $y$ represents a nilpotent class whose $k+ 1$-power is $0$. Therefore, $y^{k+1}$ must lie in the image of a differential, and (3) follows.

Proof. The statement follows from the fact that the composite being null-homotopic implies that $x$ lifts in the tower of fibrations to $F^{S^{1}}_{tr}(S({{\mathbb {C}}}^{k+1})\times X_+,\, E)^{S^{1}}$.

Proof. We observe that $E$ is connective implies that $\Sigma F({{\mathbb {C}}}P^{k},\, E)$ is $(-2k + 1)$-connective (that is, the homotopy groups are $0$ in degree $\leq -2k$). Therefore, the composite

\[ S^{{-}n} \stackrel{y}{\to} \Sigma F({{\mathbb{C}}}P^{\infty},E) \to \Sigma F({{\mathbb{C}}}P^{k},E) \]

is trivial for degree reasons if $-n\leq -2k$. From the commutative square

we deduce that the composite in the lower row is trivial. Hence, from proposition 2.8 we get that $d_r(x)=0$ if $r\leq n$.

Via the map $q_H : E \to H\pi _0 E$, we observe that (2.1) also holds for $q_H(x)$ when we replace $E$ by $H\pi _0 E$. Therefore, in the associated spectral sequence $d_r^{H}(q_H(x))=0$ if $r\leq n$ and $d_{n+1}^{H}(q_H(x))$ is described in the formula in example 2.9. Also we need to only assume $n$ is odd, as the result is vacuously true in the other case. We fix $k=\frac {n+1}{2}$ so that $n=2k-1$. With this choice of $n$ and $k$,

\[ \pi_{{-}n}\Sigma F({{\mathbb{C}}}P^{k}, E) \cong \pi_0 E, ~\pi_{{-}n} \Sigma F({{\mathbb{C}}}P^{k-1}, E)=0. \]

It follows that the composite of $y$ to $\Sigma F({{\mathbb {C}}}P^{k},\, E)$ lifts to $y_k: S^{-n} \to \Sigma F({{\mathbb {C}}}P^{k}/{{\mathbb {C}}}P^{k-1},\,E)$.

The differential $d_{n+1}(x)$ may be described as the composite

\[ S^{{-}n} \stackrel{\chi}{\to} F^{S^{1}}_{tr} (S({{\mathbb{C}}}^{k})\times X_+, E)^{S^{1}} \to \Sigma F^{S^{1}}_{tr} ([S({{\mathbb{C}}}^{k+1})/S({{\mathbb{C}}}^{k})]\wedge X_+, E)^{S^{1}} \]

where $\chi$ is a lift of $x$ along the map $F^{S^{1}}_{tr} (S({{\mathbb {C}}}^{k})\times X_+,\, E)^{S^{1}} \to F(X_+,\,E)$. We expand this in the diagram below

Observe that the bottom square is a homotopy pullback square of spectra as the homotopy fibre of both the vertical maps are $F^{S^{1}}_{tr}(Q(k-1),\, E)^{S^{1}}$. Therefore, the map $\chi$ is determined from $x$ and the map $S^{-n} \to \Sigma F^{S^{1}}_{tr} ([S({{\mathbb {C}}}^{\infty })/S({{\mathbb {C}}}^{k})]\wedge X_+,\, E)^{S^{1}}$. This may now be computed using the lift of $y$ to $\Sigma F ({{\mathbb {C}}}P^{\infty }/{{\mathbb {C}}}P^{k-1},\, E)^{S^{1}}$ as its restriction to ${{\mathbb {C}}}P^{k-1}$ is $0$. We may now compute $d_{n+1}(x)$ via the following commutative diagram

The middle vertical map is the one which quotients out the factor $X$, and this also induces the right vertical map. Under the identification $\Sigma F({{\mathbb {C}}}P^{k}/{{\mathbb {C}}}P^{k-1},\,E) \simeq \Sigma ^{-2k+1} E$, and $\pi _{-n} \Sigma ^{-2k+1} E \cong \pi _0 E$, we identify $y_k$ with $d_{n+1}^{H}(x)$.

Proof. We prove the results by induction on $k$, constructing the generators $z_i$ along the way. For $k=1$, the Stiefel manifold is the sphere $S^{2n-1}$, and in this case, we know that the $E$-cohomology is the exterior algebra on one generator. This starts the induction.

In the induction step, we know that $E^{\ast }(W_{n-1,k-1})$ is as described in this proposition, and attempt to derive the same for $E^{\ast }(W_{n,k})$ via the pushout (3.1). This gives us the following maps between long exact sequences corresponding to the columns of (3.1).

(3.2)

We now justify the various identifications described in (3.2). The fact that $E$ is complex oriented implies $E^{\ast }({{\mathbb {C}}}P^{n-1}) \to E^{\ast } ({{\mathbb {C}}}P^{n-2})$ is surjective, and the induction hypothesis gives us that $E^{\ast }(W_{n-1,k-1})$ is a free $E^{\ast }(pt)$-module. This implies that

\[ E^{r}(\Sigma({{\mathbb{C}}}P^{n-1}_+)\times W_{n-1,k-1}) \to E^{r}(\Sigma({{\mathbb{C}}}P^{n-2}_+)\times W_{n-1,k-1}) \]

is surjective. This implies the identifications in the bottom row of (3.2). Note that the map $\Sigma ({{\mathbb {C}}}P^{n-2}_+)\times W_{n-1,k-1} \to W_{n-1,k-1}$ has a section corresponding to the inclusion of the base-point of $\Sigma ({{\mathbb {C}}}P^{n-2}_+)$, and thus, the map induced on $E$-cohomology is injective. The identifications on the top follow from the ones of the bottom row, and the fact that $E^{r} (W_{n-1,k-1}) \to E^{r}(\Sigma ({{\mathbb {C}}}P^{n-2}_+)\times W_{n-1,k-1})$ is injective. It follows that we have short exact sequences

\[ 0 \to E^{{\ast}} (\Sigma^{2n-1} {W_{n-1, k-1}}_+) \stackrel{j^{{\ast}}}{\to} E^{{\ast}}(W_{n,k}) \stackrel{i^{{\ast}}}{\to} E^{{\ast}}(W_{n-1,k-1}) \to 0. \]

For $n-k+1 \leq i \leq n-1$, we choose $z_i \in E^{2i-1}(W_{n,k})$ so that they map to $z_i$ under $i^{\ast }$. The class $z_n$ is chosen so that it maps to $\Sigma x^{n-1}$ under $\mu _{n,n-1}$. From (3.2), it follows that $z_n$ is a generator for the ideal of $E^{\ast }(W_{n,k})$ given by image of $j^{\ast }$. By the construction (1) and (2) follow. As the classes $z_i$ are in odd degree and $E^{\ast }(pt)$ has no $2$-torsion, we have $z_i^{2}=0$, and (3.2) implies that $E^{\ast }(W_{n,k})$ additively matches with the exterior algebra on the $z_i$. The result now follows by induction on $k$.

Proof. We note that using proposition 3.1, it suffices to prove the statements about $\tau _j$. Consider the following commutative diagram

(3.3)

We have $\lambda (c^{BP}_j)=c_j$, and also that $\lambda$ maps the complex orientation of $BP$ to that of $H$. It readily follows that $\phi ^{*}_{BP}(c^{BP}_j) - \Sigma ^{2}x^{j-1}$ lies in the kernel of $\lambda$, which is the ideal $(v_1,\,v_2,\,\ldots )$. As

\[ \widetilde{BP}^{{\ast}} (\Sigma^{2} ({{\mathbb{C}}}P^{n-1}_+)) \cong {{\mathbb{Z}}}_{(p)}[v_1,v_2,\ldots ]\{\Sigma^{2} 1, \Sigma^{2} x, \ldots, \Sigma^{2} x^{n-1}\}, \]

the left vertical arrow of (3.3) is an isomorphism in degree $2n$. It follows that $\phi _{BP}^{\ast }(c_n^{BP}) = \Sigma ^{2} x^{n-1}$, and so, $\tau _n^{BP}=c_n^{BP}$ maps to the element of $BP^{\ast }(\Sigma ^{2} ({{\mathbb {C}}}P^{n-1}_+))$ required by the proposition.

We proceed to construct the $\tau _j$ such that $\phi _{BP}^{\ast } \tau _j=\Sigma ^{2} x^{j-1}$. Starting from $j=n$, suppose that $\tau _{j+1}$ has already been defined. We now have $\phi _{BP}^{\ast } (c_j^{BP})-\Sigma ^{2} x^{j-1} \in (v_1,\,v_2,\,\ldots )$. For degree reasons we have

\[ \phi_{BP}^{{\ast}} (c_j^{BP})-\Sigma^{2} x^{j-1} = \sum_{k > j} \rho_k \Sigma^{2} x^{k-1}=\sum_{k>j} \rho_k \phi_{BP}^{{\ast}} (\tau_k), \]

for some $\rho _k \in (v_1,\,v_2,\,\ldots )$. Rearranging terms and substituting the formula for $\tau _k$, we obtain an equation

\[ \Sigma^{2} x^{j-1} = \phi_{BP}^{{\ast}}(c_j^{BP} + \sum_{k>j} \nu_k c_k^{BP}), \]

where $\nu _k$ is a $BP^{\ast }(pt)$-linear combination of the classes $\rho _k$ in the preceeding equation. It follows that $\tau _j=c_j^{BP} + \sum _{k>j} \nu _k c_k^{BP}$ satisfies the required criteria. We note that $\phi _{BP}^{\ast }$ has image in $BP^{\ast } (\Sigma ^{2} ({{\mathbb {C}}}P^{n-1}_+))$ which is a suspension. It follows that the decomposable elements over $BP^{\ast }(pt)$ map to $0$ under $\phi _{BP}^{\ast }$. Also the formula $\phi _{BP}^{\ast }(\tau _j)=\Sigma ^{2} x^{j-1}$ implies that $\phi _{BP}^{\ast }$ induces an isomorphism when restricted to the module of indecomposables. This shows that the elements $\nu _k$ are unique, and so the classes $\tau _k$ are coherently defined over $n$ as required in the proposition.

Proof. We have a diagram of fibrations

which induces the commutative diagram

(3.4)

From the construction of the classes $y_j^{BP}$ we have $\pi ^{\ast } \alpha (\tau _j)=\delta (y_j^{BP})$. On the other hand, if $j>n-k$, the class $\alpha (\tau _j)$ maps to $0$ in $BP^{\ast }(BU(n-k))$. The map $BP^{\ast } BU(n)\to BP^{\ast } BU(n-k)$ is the map on $BP$-cohomology associated to the standard inclusion $BU(n-k)\to BU(n)$ classifying the sum of the canonical bundle with $k$-copies of a trivial bundle. This maps $c_j^{BP}$ to $0$ for $j>k$, and hence, from the formula in proposition 3.2, the classes $\tau _j$ to $0$ if $j>k$. It follows that for $j>k$, there are classes $y_j \in BP^{\ast }(W_{n,k})$ such that $\alpha (\tau _j)= \delta (y_j)$, and from (3.4) that $\pi ^{\ast }(y_j)= y_j^{BP}$. Also the property $\phi _{BP}^{\ast }(\tau _j)=\Sigma ^{2} x^{j-1}$ implies that the classes $y_j$ satisfy (2) of proposition 3.1. This result follows readily.

Proof. The proof will follow from the existence of a diagram as in (2.1). We have the commutative diagram

in which the rows are fibrations [Reference Astey, Gitler, Micha and Pastor3]. The map $f$ classifies the $n$-fold Whitney sum of universal canonical complex line bundle. It leads to the following commutative diagram

Suppose that the class $\tau _j$ of proposition 3.2 is mapped to $\psi _j$ under $f^{*}$. In the first row, the image of $y_j$ and image of $\tau _j$ coincide (proposition 3.2 and (3.4)), hence, the same must happen in the bottom row leading to the following homotopy commutative diagram

(4.2)

The Whitney sum formula for Chern classes over $BP$-cohomology implies that $f^{\ast } (c_j^{BP})= \binom {n}{j} x^{j}$. The description of $\tau _j$ in proposition 3.2 now leads to the following form for $\psi _j$

(4.3)\begin{equation} \psi_j = \binom{n}{j}x^{j} + \sum_{k>j}\nu_k\binom{n}{k}x^{k} . \end{equation}

Now apply proposition 2.10 to get $d_r(y_j)=0$ if $r<2j$, and $d_{2j}y_j$ is determined from the corresponding spectral sequence over $H{{\mathbb {Z}}}_{(p)}$. This may be computed as in [Reference Astey, Gitler, Micha and Pastor3] to be $d_{2j} y_j= \binom {n}{j}x^{j}$. Hence the result follows.

Proof. The class $x$ is a permanent cycle by proposition 2.6. The multiplicative structure determines all the differentials once they are known on the classes $y_j$. We notice that $E_{2n+1}$ is the $E_\infty$-page because $d_{2n}(y_n)= x^{n}$ (proposition 4.1) and so all the higher powers of $x$ are killed in the $E_{2n}$-page.

From proposition 4.1, we see that the first non-trivial differential is $d_{2(n-k+1)}$ and the generator $y_{n-k+1}$ and all its multiples do not survive to the next page since

\[ d_{2(n-k+1)}(y_{n-k+1})=\binom{n}{n-k+1}x^{n-k+1}. \]

For the $y_j$ of higher degree, it may happen that the first non-trivial differential on it

\[ d_{2j}y_j = \binom{n}{j}x^{n-k+1} \]

may be zero. This precisely happens when $\binom {n}{j}$ lies in the ideal generated by $\binom {n}{i}$ for $n-k+1\leq i < j$ inside ${{\mathbb {Z}}}_{(p)}$. This condition may be interpreted in terms of $p$-adic valuations of these numbers. We then obtain a multiple $p^{s} y_j$ on which the differential is $0$, determined by the formula $s+v_p(\binom {n}{j})= \min _{n-k+1\leq i < j}v_p(\binom {n}{i})$. The class $p^{s} y_j$ may now support higher order differentials. Their formula is determined by computing $p^{s}\psi _j$ using (4.2) in the form of (2.1)

for $N> 2j$. According to the formula (4.3), the next possible differential is

\[ d_{2j+2}(p^{s} y_j)= p^{s}\nu_{j+1} \binom{n}{j+1}x^{j+1} = p^{s} \nu_{j+1} d_{2j+2}(y_{j+1}). \]

We now rectify this class as $p^{s}y_j - p^{s} \nu _{j+1} y_{j+1}$ and obtain a cycle. This process continues until we reach the $E_{2n+1}$-page following which there are no further non-zero differentials.

We now formalize the above process by writing down a series of modifications to produce the element $\gamma _j$. Starting with $\gamma _{j}^{(2)}:= y_{j}$, in $r$-th step of the modification, the modified element will be denoted by $\gamma _{j}^{(r)}$. Below we describe transformations, exactly one of which will be performed to produce $\gamma _{j}^{(r+1)}$ from $\gamma _{j}^{(r)}$.

1. (1) If $d_r\gamma _{j}^{(r)}=0$, then it survives to the next page and we call that element $\gamma _{j}^{(r+1)}$.

2. (2) If $r=2j$ and $\gamma _{j}^{(2j)}=y_{j}$ and $d_r(\gamma _{j}^{(r)}) = \binom {n}{j} x^{j}$. Define $s$ by the formula $s+v_p(\binom {n}{j})= \min _{n-k+1\leq i < j}v_p(\binom {n}{i})$, and declare $\gamma _{j}^{(r+1)}=p^{s} \gamma _{j}^{(r)}$.

3. (3) If $r>2j$, and $d_r(\gamma _j^{(r)})\neq 0$, then we know $r$ is even, and there is a $BP^{\ast }$-multiple of $y_{\frac {r}{2}}$ mapped by $d_r$ onto the same class (this follows from the formula for $\psi _j$ in the same way as demonstrated for $p^{s}y_j$ above). That is, $d_r(\gamma _j^{(r)})= \lambda d_{r}(y_{\frac {r}{2}})$ for some $\lambda \in BP^{\ast }(pt)$. We declare $\gamma _{j}^{(r+1)}=\gamma _{j}^{(r)}-\lambda y_{\frac {r}{2}}$.

We finally write $\gamma _j = \gamma _j^{(2n+1)}$ which survives to the $E_\infty$-page. Hence, we have shown that the $0$-th column of the $E_\infty$-page is $\Lambda _{BP^{*}}(\gamma _{n-k+2},\,\ldots,\,\gamma _n)$. Also on the $E_\infty$-page the ideal generated by $\{\binom {n}{j}x^{j}\vert n-k< j\leq n \}$ goes to $0$, as each of the generators are hit by the differentials $d_{2j}(y_j)$. This completes the proof.

Proof. The proof goes by induction on $k$. For $k=1$, the fibration is up to homotopy the following sequence

so that the kernel of $p_1^{*}$ is the ideal generated by $x^{n}$, satisfying the statement of the lemma. Suppose that the lemma is true for $PW_{n,k-1}$. To show the result for $PW_{n,k}$, we consider the diagram

In the above diagram, $f$ classifies the bundle $n\gamma$ where $\gamma$ is the canonical line bundle over ${{\mathbb {C}}}P^{\infty }$, and $q$ is induced by the $S^{1}$-equivariant projection $W_{n,k}\to W_{n,k-1}$. The three squares in the diagram are homotopy pullbacks. Our aim is to understand the kernel of $q^{*}$. We see that $PW_{n,k}\to PW_{n,k-1}$ is, up to homotopy, the sphere bundle associated to the complex bundle classified by the map $T_{n,k-1}$. This is because $BU(n-k)$ is (up to homotopy) the sphere bundle of the canonical $n-k+1$-plane bundle over $BU(n-k+1)$. As $BP$ is complex oriented, we obtain a Gysin sequence

It follows that the kernel of $q^{\ast }$ is the ideal generated by $e^{BP}(T_{n,k-1})$ in $BP^{\ast } PW_{n,k-1}$. The bundle $T_{n,k-1}$ is obtained by lifting the composite $PW_{n,k-1} \stackrel {p_{k-1}}{\to } {{\mathbb {C}}}P^{\infty } \stackrel {n\gamma }{\to } BU(n)$ to $BU(n-k)$ so that $T_{n,k-1} + (k-1)\epsilon = np_{k-1}^{\ast } \gamma$. We readily compute $e^{BP}$ as the top $BP$-Chern class

\[ e^{BP}(T_{n,k-1})=p_{k-1}^{{\ast}} c_{n-k+1}(n\gamma)=\binom{n}{n-k+1} x^{n-k+1}. \]

Therefore, $\binom {n}{n-k+1}x^{n-k+1}$ lies in the kernel of $p_k^{\ast }: BP^{\ast } {{\mathbb {C}}}P^{\infty } \to BP^{\ast } PW_{n,k}$. By the inductive formula for the kernel of $p_{k-1}^{\ast } : BP^{\ast } {{\mathbb {C}}}P^{\infty } \to BP^{\ast } PW_{n,k-1}$, the proof is now complete.

Proof. Lemma 4.3 implies that $p^{\ast }$ induces a ring map of $BP^{\ast }$-modules $BP^{\ast }(pt)[[x]]/I \to BP^{\ast }(PW_{n,k})$. Choosing representatives for generators $\gamma _j$ of proposition 4.2 in the $E_\infty$-page we obtain a $BP^{\ast }(pt)$-module map $\Lambda _{BP^{*}(pt)}(\gamma _{n-k+2},\,\ldots,\,\gamma _n)$ to $BP^{\ast } PW_{n,k}$. The multiplication as a bilinear map on these factors gives a map

\[ \Lambda_{BP^{*}(pt)}(\gamma_{n-k+2},\ldots,\gamma_n)\otimes_{BP^{*}(pt)} BP^{*}(pt)[[x]]/I \to BP^{{\ast}} PW_{n,k} \]

of $BP^{\ast }(pt)$-modules. This is an isomorphism by proposition 4.2 and the multiplicative structure of the spectral sequence (4.1). Further if $p\neq 2$, we have $\gamma _j^{2}=0$ as $\gamma _j$ lies in odd degree. Therefore, the isomorphism is also multiplicative.