MATHEMATICS

A. S. Schwarz

ON SOME CONCEPTS RELATED TO THE NOTION OF THE GENUS OF A FIBERED SPACE

(Presented by Academician P. S. Aleksandrov, 7 VII 1960)

We shall use the definitions and notation of the notes \((^{1-3})\). Let \(G\) be a topological group. We regard the space of the group \(G\) as a \(G\)-space, assuming that the group \(G\) acts on it by left translations. The join \(G * \cdots * G\) of \(k\) copies of the \(G\)-space \(G\) will be denoted by \(A_k\), and the orbit space \((A_k)_G\) of the group \(G\) in \(A_k\) by \(B_k\). By the cone over a \(G\)-space \(X\) we shall mean the \(G\)-space \(X * G\).

The genus of a \(G\)-space \(X\) (i.e., the genus of the fibration \(X \to X_G\)) will be denoted by \(g(X)\); by virtue of Theorem 5 of note \((^1)\), the genus of the \(G\)-space \(X\) can be defined as the least \(k\) for which there exists an admissible mapping of the \(G\)-space \(X\) into the \(G\)-space \(A_k\). The homological genus of the \(G\)-space \(X\) will be denoted by \(h(X)\).

It is easy to see that the genus of the cone \(X * G\) over the \(G\)-space \(X\) is either equal to the genus of the \(G\)-space \(X\), or exceeds it by one:

\[ g(X) \leq g(X * G) \leq g(X) + 1. \]

This remark suggests the following.

Definition 1. The \(S\)-genus of a \(G\)-space \(X\) (denoted by \(Sg(X)\)) will be called the least of the numbers \(k\) for which

\[ g(X * A_l) \leq k + l \]

for all sufficiently large \(l\).

It is obvious that \(Sg(X) \leq g(X)\). It can be proved that in the case when the group \(G\) is finite,

\[ h(X * G) = h(X) + 1 \]

and, consequently, in this case

\[ h(X) \leq Sg(X). \]

Let us note the following proposition:

If \(f: X \to Y\) is an admissible mapping of the \(G\)-space \(X\) into the \(G\)-space \(Y\) and \(Sg(X)=Sg(Y)\), then the mapping \(f\) is not homotopic to zero.

If the group \(G\) is finite, then in this assertion the condition \(Sg(X)=Sg(Y)\) can be replaced by the condition \(h(X)=h(Y)\).

M. A. Krasnosel’skii posed in the book \((^5)\) a question which can be expressed in the notation adopted here as follows: if \(G=Z_2\) is the group of order two, then for every \(G\)-space \(X\) does the equality \(Sg(X)=g(X)\) hold (in other words, is it always true that \(g(X * Z_2)=g(X)+1\))? We give an example which yields a negative answer to this question.

To each mapping \(\varphi: S^m \to S^n\) of the \(m\)-dimensional sphere into the \(n\)-dimensional sphere, let us associate a \(Z_2\)-space \(E_\varphi\) by means of the following construction: the space \(E_\varphi\) is obtained from the sphere \(S^n\) by attaching two \((m+1)\)-dimensional balls, the first of which is attached by means of the mapping \(\varphi\), and the second by means of the mapping \(T\varphi\) (\(T\) is the central symmetry of the sphere \(S^n\)); the unique nontrivial element of the group \(Z_2\) acts on the sphere \(S^n \subset E_\varphi\) as the central symmetry \(T\), and interchanges the attached balls. If \(\alpha \in \pi_m(S^n)\), \(\varphi: S^m \to S^n\) is a mapping of the class \(\alpha\), then the genus of the \(Z_2\)-space \(E_\varphi\) does not depend on the choice of the mapping \(\varphi\), and therefore we may introduce the notation

\[ g(\alpha)=g(E_\varphi). \]

The following assertion holds:

If \(\alpha \in \pi_m(S^n)\), where \(n \leqslant 3\), \(n=7\), or \(m<2n-1\), then \(g(\alpha)=n+1\) in the case when the element \(\alpha\) has odd order, and \(g(\alpha)=n+2\) otherwise.

It is not difficult to verify that \(E_\varphi * Z_2 = ES\varphi\), where \(S\varphi\) is the suspension over the mapping \(\varphi\); therefore, in order to refute Krasnosel’skii’s conjecture, it is enough to construct an element \(\alpha \in \pi_m(S^n)\) such that \(g(S\alpha)\ne g(\alpha)+1\). But from the proposition formulated above it follows that for any element \(\alpha \in \pi_3(S^2)\) having even Hopf invariant, one has \(g(\alpha)=g(S\alpha)=4\).

Definition 2. The dual genus of a \(G\)-space \(X\) (denoted by \(\tilde g(X)\)) is the greatest of the numbers \(k\) for which there exists an admissible mapping of the \(G\)-space \(A_k\) into \(X\).

Definition 3. The dual homological genus of a \(G\)-space \(X\) (denoted by \(\tilde h(X)\)) is the least of the numbers \(k\) for which there is a cohomology class \(z\) of the classifying space \(B_G\) of the group \(G\) (with coefficients in a local system) satisfying the conditions \(\varphi^*(z)=0\), \(\varphi_{k+1}^*(z)\ne 0\) (here \(\varphi\), \(\varphi_{k+1}\) denote the characteristic mappings \(X_G\) into \(B_G\) and \(B_{k+1}=(A_{k+1})_G\) into \(B_G\)). If the group \(G\) is discrete, then the condition \(\varphi_{k+1}^*(z)\ne 0\) appearing in this definition may be replaced by the condition \(\dim z \leqslant k\).

It is easy to see that \(\tilde g(X)\leqslant \tilde h(X)\).

Let \(X\) be a \(G\)-space, and let \(B\to B_G\) be the universal fibration with group \(G\). Denote by \(X'_G\) the trajectory space \((X\times B)_G\) of the group \(G\), acting coordinatewise in \(X\times B\); this space is weakly homotopy equivalent to the space \(X_G\). The projection \(X\times B\to B\) gives rise to a mapping \(p:X'_G=(X\times B)_G\to B_G\). The mapping \(p\) is a fibration with fibre \(X\); this fibration will be denoted by \(\mathfrak B(X'_G,B_G,X,p)\) (see \((^4)\), p. 209).

The problem of computing the dual genus can be reduced to the problem of the possibility of constructing a section by means of the following theorem:

Let \(X\) be a \(G\)-polyhedron, \(G\) a Lie group (not necessarily connected), and let \(\mathfrak B'(Z,B_k,X,p')\) be the fibration over \(B_k\) induced from the fibration \(\mathfrak B\) by means of the mapping \(\varphi_k:B_k\to B_G\) (the characteristic mapping of the fibration \(A_k\to B_k\)). Then \(\tilde g(X)\geqslant k\) if and only if the fibration \(\mathfrak B'\) has a section.

Applying the theorem on the first obstruction to the extension of a section, we obtain the following assertion:

Let the space of the group \(G\) be a \(d\)-dimensional polyhedron, and let \(X\) be a \(G\)-polyhedron that is aspherical in dimensions \(<s\) \((s\geqslant 2)\). If \(s\geqslant (d+1)(k-1)\), then \(\tilde g(X)\geqslant k\); if \(s=(d+1)(k-1)-1\), then \(\tilde g(X)=k-1\) if and only if \(\tilde h(X)=k-1\).

Let \(G\) be a discrete group, \(G=S^1\) or \(G=S^3\) (in these cases the space of the group \(G\) is aspherical in dimensions \(<d\), where \(d=\dim G\) is the dimension of this space). Then the dual genus of the \(G\)-polyhedron \(X\geqslant k\) if and only if, over the \((d+1)(k-1)\)-dimensional skeleton of the base \(B_G\) of the fibration \(\mathfrak B(X'_G,B_G,X,p)\), there exists a section.

Using this assertion and applying known results on the second obstruction to the extension of a section \((^6,^7)\), one can give conditions, expressed in terms of the homological properties of the spaces \(X\) and \(X_G\), for the equality \(\tilde g(X)=k-1\) to hold, if the space \(X\) is aspherical in dimensions
\[ <s=(d+1)(k-1)-2. \]
We give here only the result for \(G=S^3\), which has an especially simple form.

Let \(G=S^3\); let \(X\) be a \(G\)-space aspherical in dimensions
\[ <s=4(k-1)-2; \]
\(b\) an element of the group \(H^s(X_G,\pi_s(X))\) which maps

under the homomorphism \(\pi^*: H^s(X_G,\pi_s(X)) \to H^s(X,\pi_s(X))\), generated by the identification map \(\pi: X \to X_G\), into the fundamental class of the space \(X\). Then \(\bar g(X)=k-1\) if and only if the class \(Sq^2 b \in H^{s+2}(X_G,\pi_{s+1}(X))\) is nonzero.

Definition 4. The dual \(S\)-genus of a \(G\)-space \(X\) (denoted \(S\bar g(X)\)) is the largest of the numbers \(k\) for which \(\bar g(X*A_l)\ge k+l\) for some \(l\).

It is obvious that \(S\bar g(X)\ge \bar g(X)\). If \(G\) is a finite group, then one can show that \(S\bar g(X)\le \bar h(X)\).

Let us note that the \(S\)-genus and the dual \(S\)-genus can be simply characterized by means of the notion of an \(R\)-mapping: if \(R\) is a \(G\)-space into which every \(G\)-space of genus \(\le g(R)\) can be admissibly mapped (for example, if \(R=G\)), then \(Sg(X)\le k\) \([S\bar g(X)\ge k]\) if and only if there exists an \(R\)-mapping of the \(G\)-space \(X\) into the \(G\)-space \(A_k\) (of the \(G\)-space \(A_k\) into the \(G\)-space \(X\)).

The \(S\)-genus and the dual \(S\)-genus do not decrease under \(R\)-mappings; more precisely, if the \(G\)-space \(R\) satisfies the conditions imposed above and there exists an \(R\)-mapping of the \(G\)-space \(X\) into the \(G\)-space \(Y\), then \(Sg(X)\le Sg(Y)\) and \(S\bar g(X)\le S\bar g(Y)\). If the group \(G\) is finite, then under the same conditions \(h(X)\le h(Y)\) and \(\bar h(X)\le \bar h(Y)\).

Let \(G\) be a finite group; let \(R\) be an \(r\)-dimensional \(G\)-polyhedron homotopy equivalent to the \(r\)-dimensional sphere (such a \(G\)-polyhedron exists if and only if every abelian subgroup of the group \(G\) is cyclic \((^8)\)).

Suppose that \(X\) is a \(G\)-polyhedron; \(Y=D_nX\) is the \(G\)-polyhedron \(n\)-dual to the \(G\)-polyhedron \(X\) with respect to \(R\). Then:

1. \(Sg(X)+S\bar g(Y)=(r+1)n\).
2. \(h(X)+\bar h(Y)=(r+1)n\).
3. If \(\dim X\le 2Sg(X)-3\), then \(g(X)=Sg(X)\).
4. If the polyhedron \(X\) is aspherical in dimensions \(<\frac12 S\bar g(X)\), then \(\bar g(X)=S\bar g(X)\).

Analogous assertions can be proved in the case when \(G=S^1\) or \(G=S^3\).

As is clear from the theorems formulated above, the dual genus is easier to compute than the genus of a \(G\)-space; therefore the relation
\[ Sg(X)=(r+1)n-S\bar g(D_nX) \]
may perhaps be used to compute the genus of concrete \(G\)-polyhedra.

Voronezh State University

Received
4 VII 1960

REFERENCES

1. A. S. Schwarz, DAN, 119, No. 2, 219 (1958).
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3. A. S. Schwarz, DAN, 136, No. 1 (1961).
4. A. Borel, Seminar on Fiber Spaces, IL, 1958, p. 163.
5. M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.
6. V. G. Boltyanskii, Izv. Akad. Nauk SSSR, Ser. Mat., 20, 99 (1956).
7. A. M. Vinogradov, DAN, 130, No. 4 (1960).
8. R. G. Swan, Bull. Am. Math. Soc., 65, No. 6, 368 (1959).