Mathematics

A. Matuzya­vichus

Secant Surfaces of Double Fibrations

(Presented by Academician P. S. Aleksandrov on 20 X 1956)

Let (\mathfrak{F}_1={P_1,p,B}) and (\mathfrak{F}_2={P_2,p',P_1}) be two fiber spaces in the sense of Serre (1), the base of (\mathfrak{F}_1) being a simply connected simplicial complex (B). The fibers (C_1=p^{-1}(x_0)) and (C'=(p')^{-1}()) (where (x_0\in B,\ \in C_1\subset P_1)) are assumed to be homotopically simple in dimensions (r) and ((r-1)), respectively. Suppose that over an (r)-dimensional skeleton (B^r) of the base space (B), two secant surfaces (\mathfrak{S}_1) and (\mathfrak{S}_2) are given in (\mathfrak{F}_1), coinciding over the skeleton (B^{r-1}).

The mapping (\mathfrak{S}i:B^r\to P_1) ((i=1,2)) and the fiber space (\mathfrak{F}_2) induce the fiber space
[
\mathfrak{F}i}={Pi},p_i,B^r}
]
with base (B^r). Here (P_i(x)=p'(g)), and the projection (p_i) is defined by the formula (p_i(x,g)=x).}_i}) is the subspace of the direct product (B^r\times P_2) consisting of all pairs ((x,g)) satisfying the condition (\mathfrak{S

Since (\mathfrak{F}{\mathfrak{S}_1}) and (\mathfrak{F}2}) coincide over (B^{r-1}), any secant surface (\psi), given in (\mathfrak{F}1}) on (B^{r-1}), may also be regarded as a secant surface in (\mathfrak{F}2}) on (B^{r-1}). We shall assume the fiber spaces (\mathfrak{F}_1,\mathfrak{F}_2) and the secant surfaces (\mathfrak{S}_1,\mathfrak{S}_2) to be such that, in the fiber spaces (\mathfrak{F}1}) and (\mathfrak{F})) and henceforth denote it by (\psi).}_2}), a secant surface can be constructed over the entire ((r-1))-dimensional skeleton (B^{r-1}). Fix one such secant surface (on (B^{r-1

Denote by (z^r_{1,\psi}) and (z^r_{2,\psi}) the obstructions to extending the secant surface (\psi) in the fiber spaces (\mathfrak{F}{\mathfrak{S}_1}) and (\mathfrak{F}(C'))).}_2}), and by (Z^r_1) and (Z^r_2) the cohomology classes* of these obstructions. Here (Z^r_1) and (Z^r_2) are elements of the cohomology group (H^r(B^r,\pi^{r-1

Next denote by (d^r_{\mathfrak{S}1,\mathfrak{S}_2}) the difference cochain of the secant surfaces (\mathfrak{S}_1) and (\mathfrak{S}_2). We may regard this cochain as an (r)-dimensional cocycle of the complex (B^r). The cohomology class of this cocycle shall be denoted by (D^r); this class is an element of the group}_1,\mathfrak{S}_2
[
H^r(B^r,\pi^r(C_1)).
]

Finally, put (C_2=(p')^{-1}(C_1)) and denote by (p') again the mapping (p') considered on (C_2). Then ({C_2,p',C_1}) is a fiber space (a part of the fiber space (\mathfrak{F}_2)) with fiber (C'=(p')^{-1}(*)).

* Following a proposal of V. G. Boltyansky, we shall use the term cohomology instead of the previously used upper homology, (\nabla)-homology, cohomology, since this term better corresponds to the nature of the concept under consideration. Accordingly, we shall speak of cochains, cocycles, etc.

For this fiber space we can write the exact homotopy sequence

[
\ldots \longrightarrow \pi^{r}(C_{2}) \xrightarrow{p'*} \pi^{r}(C)
\xrightarrow{\Delta} \pi^{r-1}(C') \longrightarrow \pi^{r-1}(C_{2}) \longrightarrow \ldots ,
\tag{1}
]

where (\Delta) is the boundary homomorphism. This homomorphism

[
\Delta:\quad \pi^{r}(C_{1}) \longrightarrow \pi^{r-1}(C')
]

induces a homomorphism of cohomology groups

[
H^{r}\bigl(B^{r},\pi^{r}(C_{1})\bigr)\longrightarrow
H^{r}\bigl(B^{r},\pi^{r-1}(C')\bigr),
]

which we shall denote by (\hat\Delta).

Under the stated assumptions the following theorem holds.

Theorem. The cohomology classes (Z_{1}^{r}, Z_{2}^{r}, D^{r}{\mathfrak S) are related by},\mathfrak S_{2}

[
Z_{1}^{r}-Z_{2}^{r}=\hat\Delta D^{r}{\mathfrak S .},\mathfrak S_{2}
\tag{2}
]

An essential role in the proof of this theorem is played by the following construction. Let (\alpha) be an element of the homotopy group (\pi^{r}(C_{1})), and let (f:S^{r}\to C_{1}) be a map of the oriented sphere (S^{r}) which takes some point (y\in S^{r}) to the point (*) and belongs to the class (\alpha). The map (f) and the fiber space ({C_{2},p',C_{1}}) induce a new fiber space with base (S^{r}) and fiber (C'). Denote by (Z^{r}\in H^{r}\bigl(S^{r},\pi^{r-1}(C')\bigr)) the characteristic cochain (i.e. the cohomology class of the first obstruction) of this fiber space, and by (\beta\in \pi^{r-1}(C')) the index of the cohomology class (Z^{r}) on the oriented sphere (S^{r}). It is easily established that the element (\beta) does not depend on the choices made in the construction and is uniquely determined by the element (\alpha), so that one may set (\beta=\chi(\alpha)).

Lemma. The map defined above

[
\chi:\quad \pi^{r}(C_{1})\longrightarrow \pi^{r-1}(C')
]

coincides with the boundary homomorphism (\Delta) of the exact sequence (1).

Let us outline, in its main features, the proof of formula (2). Let (T^{r}) be an arbitrary (r)-dimensional oriented simplex of the complex (B), and let (T^{+}) and (T^{-}) be two identical copies of it, glued along their boundaries. We orient the sphere (S^{r}=T^{+}\cup T^{-}) consistently with (T^{+}). The map

[
\mathfrak S(x)=
\begin{cases}
\mathfrak S_{1}(x), & \text{if } x\in T^{+},\
\mathfrak S_{2}(x), & \text{if } x\in T^{-}
\end{cases}
]

of the sphere (S^{r}) into (P_{1}) and the space (\mathfrak P_{2}) induce over (S^{r}) a fiber space (\mathfrak P_{\mathfrak S}={P_{\mathfrak S},p,S^{r}}), which, evidently, on (T^{+}) coincides with the part of the space (\mathfrak P_{\mathfrak S_{1}}) over (T^{r}), and on (T^{-}) with the part of (\mathfrak P_{\mathfrak S_{2}}) over (T^{r}).

Let (k'{t}) be a deformation joining the identity map (k'\to B) with the map (k'}:T^{r{1}) that sends (T^{r}) to the point (x=k'}). Put (k_{t{t}\circ e), where (e:S^{r}\to T^{r}) maps each of the simplexes (T^{+},T^{-}) identically onto (T^{r}). From (p\circ\mathfrak S=e=k}), applying the covering homotopy existence condition to the fiber space (\mathfrak P_{1}), we find a deformation (\mathfrak S^{t}) of the map (\mathfrak S^{0}=\mathfrak S) such that (p\circ\mathfrak S^{t}=k_{t}). The map (\mathfrak S^{1}) carries the sphere (S^{r}) into the fiber (C_{1}) and determines an element (d^{r{\mathfrak S},\mathfrak S_{2}}(T^{r})) of the group (\pi^{r}(C_{1})), i.e. the value of the difference (d^{r{\mathfrak S).},\mathfrak S_{2}}) on the simplex (T^{r

The map (\mathfrak S^t:S^r\to P_1) and the fibered space (\mathfrak P_2) induce a new fibered space
[
\mathfrak P_{\mathfrak S^t}={P_{\mathfrak S^t},p_t',S^r};
]
the points of the space (P_{\mathfrak S^t}) are pairs ((x,g)), (x\in S^r,\ g\in P_2), satisfying the condition
[
\mathfrak S^t(x)=p'(g).
]
Define the map (h^t:P_{\mathfrak S^t}\to P_2) by setting
[
h^t(x,g)=g.
]

The map (\psi) is defined on the whole base (B^{r-1}) and, in particular, on the sphere (S^{r-1}=T^r). Since (\mathfrak P_{\mathfrak S_1}=\mathfrak P_{\mathfrak S_2}) over (B^{r-1}), the map (\psi), considered on (S^{r-1}), may be regarded as a section surface of the fibered space (\mathfrak P_{\mathfrak S^0}) defined on (S^{r-1}). This map
[
S^{r-1}\to P_{\mathfrak S^0}
]
will, for convenience, be denoted by (\psi^0). The map
[
\varphi^0=h^0\circ\psi^0:\quad S^{r-1}\to P_2
]
obviously satisfies the condition
[
p'\circ\varphi^0=\mathfrak S^0,
]
and, by the condition for the existence of a covering homotopy, one can find a continuous family of maps
[
\varphi^t:\quad S^{r-1}\to P_2,
]
such that (on (S^{r-1}))
[
p'\circ\varphi^t=\mathfrak S^t.
]
Now put
[
\psi^t(x)=\bigl(x,\varphi^t(x)\bigr),\quad x\in S^{r-1},
]
and we obtain a section surface (\psi^t) of the fibered space (P_{\mathfrak S^t}), defined over (S^{r-1}). In particular, we obtain the section surface
[
\psi^1:\quad S^{r-1}\to P_{\mathfrak S^1}.
]

The obstruction (in the fibered space (\mathfrak P_{\mathfrak S^1})) to extending this section surface has, on the cells (T^+) and (T^-) (oriented in the same way as (T^r)), certain values (z^+) and (z^-) (which are elements of the group (\pi^{r-1}(C')); here it is assumed that the map (\mathfrak S^1) sends some point of the sphere (S^{r-1}) to the point (*), which entails no loss of generality). According to the lemma formulated above, we have
[
z^+-z^-=\Delta d^r_{\mathfrak S_1,\mathfrak S_2}(T^r).
\tag{3}
]

It is not difficult to show (by constructing a connecting deformation depending on two parameters) that the elements (z^+) and (z^-) do not depend on the choice of the auxiliary deformations (k_t',\mathfrak S^t) and (\varphi^t), but are uniquely determined by the choice of the section surfaces (\mathfrak S_1,\mathfrak S_2,\psi) and the oriented simplex (T^r). Choosing the auxiliary deformations in a special way, it is not difficult to verify that
[
z^+=z_{1,\psi}^{\,r}(T^r),\quad z^-=z_{2,\psi}^{\,r}(T^r).
\tag{4}
]

Formula (2) follows from (3), (4).

Let us consider some special cases of the theorem proved. Let (\mathfrak P_1={P_1,p_1,B,C_1,\mathfrak G}) and (\mathfrak P_2={P_2,p_2,B,C_2,\mathfrak G}) be two skew products ((^2)) with the same coordinate transformations.

for which the base is one and the same simply connected simplicial complex (B); the fibers have the form (C_1=\mathscr G/\Gamma_1,\ C_2=\mathscr G/\Gamma_2), where (\mathscr G) is a transitive group of transformations of the fibers (C_1, C_2), and (\Gamma_1,\Gamma_2) are such stable subgroups of (\mathscr G) that the inclusion (\mathscr G\supset \Gamma_1\supset \Gamma_2) holds.

There arises a natural mapping (p') (by inclusion):

[
p':\quad C_2=\mathscr G/\Gamma_2\to C_1=G/\Gamma_1,
]

and, in consequence of the coincidence of the coordinate transformations, the mapping

[
p':\quad P_2\to P_1.
]

If the subgroup (\Gamma_2) has a local secant surface, then this mapping gives a new skew product with fiber (C'=\Gamma_1/\Gamma_2). If the same restrictions as before are imposed on the spaces (B, C_1, C'), then the conditions of applicability of the theorem proved above will be satisfied. In this case the homomorphism (\Delta) of the exact sequence (1) can be included in the following commutative diagram, which facilitates its computation:

[
\begin{array}{ccc}
\pi^r(\mathscr G,\Gamma_1) & \xrightarrow{\ p_\ } & \pi^r(\mathscr G/\Gamma_1) \[2mm]
\partial\downarrow & & \Delta\downarrow \[2mm]
\pi^{r-1}(\Gamma_1) & \xrightarrow{\ \bar p_\ } & \pi^{r-1}(\Gamma_1/\Gamma_2)
\end{array}
\quad
\begin{array}{c}
\searrow^{(p'*)^{-1}} \[4mm]
\nearrow
\end{array}
\quad
\pi^r(\mathscr G/\Gamma_2,\Gamma_1/\Gamma_2).
\tag{5}
]

Let us consider an example.

Let (\mathscr P_1={P_1,\ p_1,\ M^n,\ C_1,\ \mathscr G}), (\mathscr P'2={P_2,\ p_2,\ M^n,\ C_2,\mathscr G}) be two skew products whose base is a Riemannian (n)-dimensional manifold (M^n), oriented and triangulated, and whose spaces (P_1) and (P_2) consist of all (k)-frames and, respectively, ((k+1))-frames tangent to the manifold (M^n). The fiber (C_1) in (\mathscr P_1) is the Stiefel manifold (C_1=V=SO(n)/SO(r)) ((k=n-r)), and the fiber (C_2) in (\mathscr P'2) is (C_2=V=SO(n)/SO(r-1)). It is clear that such fibers and the group (\mathscr G=SO(n)) satisfy the required conditions.

Since (SO(n)\supset SO(r)\supset SO(r-1)), there arises a natural mapping (discarding the last vector of the frame)

[
p':\quad C_2=V_{n,k+1}\to C_1=V_{n,k}.
]

We obtain the skew product

[
\mathscr P_2={V_{n,k+1},\ p',\ V_{n,k},\ S^{r-1},\ SO(r)/SO(r-1)}.
]

In view of the triviality of the groups (\pi^s(S^{r-1})) for (s