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**Improvements of the five halves theorem of J. Boardman with respect to the decomposability degree.**
*(English)*
Zbl 1305.57051

Let \(T\) be a smooth involution on a smooth closed manifold \(M\) of dimension \(m\) and let \(F=Fix(T)\) be the set of fixed points fixed by \(T\).

Suppose that \(M\) is a non-bounding manifold, i.e., there does not exist a compact smooth manifold of dimension \(m+1\) such that \(\partial W=M\). Then J. M. Boardman [Bull. Am. Math. Soc. 73, 136–138 (1967; Zbl 0153.25403)] showed that \(m\leq (5/2).n\) where \(n\) is the maximum dimension among the components of \(F\), and moreover, this estimate is sharp. There have been many improved upper bounds for \(\dim M\) (assuming \(M\) to be non-bounding) by putting restrictions on the fixed point set \(F\). Write \(F=\cup_{0\leq k\leq n}F^k\), where \(F^k\) is the union of \(k\)-dimensional components of \(F\). C. Kosniowski and R. E. Stong [Topology 17, 309–330 (1978; Zbl 0402.57005)] showed that if \(F=F^n\), then \(m\leq 2n\). P. L. Q. Pergher and R. E. Stong [Transform. Groups 6, No. 1, 79–86 (2001; Zbl 0985.57017)] showed that if \(F=F^0\cup F^n\) then \(m\leq \mathcal{M}(n)\) where \(\mathcal{M}(n)\) defined as follows: Write \(n=2^p.q, q \text{~odd},~ p\geq 0\). Then \(\mathcal{M}(n)=2n+p-q+1\) if \(p\leq q\) and \(\mathcal{M}(n)=2n+2^{p-q}\) if \(q\geq p\).

Given a closed non-bounding manifold \(F=F^n\) of dimension \(n\), the decomposability degree of \(F\), denoted \(\ell(F^n)\), is the smallest number \(r\) such that there exists a non-dyadic partition \(\omega:=i_1,\dots, i_r\) of \(n\) with Stiefel-Whitney number \(\langle s_\omega(F),[F]\rangle \neq 0\). Here non-dyadic means that none of the \(i_t\) is of the form \(2^s-1\) and \(s_\omega(F)\) is the polynomial \(s_\omega(w_1,\dots, w_n)\in H^n(F;\mathbb{Z}_2)\) in the Stiefel-Whitney classes \(w_j=w_j(F)\) of \(F\) obtained from the monomial symmetric polynomial \(\sum x_1^{i_1}\dots x_r^{i_r}\) in the indeterminates \(x_1,\dots, x_n\) expressed as a polynomial \(s_\omega(\sigma_1,\dots, \sigma_n)\) in the elementary symmetric polynomials \(\sigma_j=\sigma_j(x_1,\dots, x_n)\).

The main result of the paper is the following: Let \(T:M\to M\) be a smooth involution of a closed \(m\)-dimensional manifold which is non-bounding and let \(F=Fix(T)\). Suppose that there exist integers \(2\leq j<n<m\) such that \(F^k=\emptyset\) if \(j<k<n\) and that the \(F^j\) is non-bounding. Then \(m\leq \mathcal{M}(n-j)+\ell(F^j)\). The authors provide examples to show, in some ‘special situations’ that the bound for \(m\) is best possible. Taking \(j=n-1\), the authors obtain an improvement on the five-halves theorem.

Suppose that \(M\) is a non-bounding manifold, i.e., there does not exist a compact smooth manifold of dimension \(m+1\) such that \(\partial W=M\). Then J. M. Boardman [Bull. Am. Math. Soc. 73, 136–138 (1967; Zbl 0153.25403)] showed that \(m\leq (5/2).n\) where \(n\) is the maximum dimension among the components of \(F\), and moreover, this estimate is sharp. There have been many improved upper bounds for \(\dim M\) (assuming \(M\) to be non-bounding) by putting restrictions on the fixed point set \(F\). Write \(F=\cup_{0\leq k\leq n}F^k\), where \(F^k\) is the union of \(k\)-dimensional components of \(F\). C. Kosniowski and R. E. Stong [Topology 17, 309–330 (1978; Zbl 0402.57005)] showed that if \(F=F^n\), then \(m\leq 2n\). P. L. Q. Pergher and R. E. Stong [Transform. Groups 6, No. 1, 79–86 (2001; Zbl 0985.57017)] showed that if \(F=F^0\cup F^n\) then \(m\leq \mathcal{M}(n)\) where \(\mathcal{M}(n)\) defined as follows: Write \(n=2^p.q, q \text{~odd},~ p\geq 0\). Then \(\mathcal{M}(n)=2n+p-q+1\) if \(p\leq q\) and \(\mathcal{M}(n)=2n+2^{p-q}\) if \(q\geq p\).

Given a closed non-bounding manifold \(F=F^n\) of dimension \(n\), the decomposability degree of \(F\), denoted \(\ell(F^n)\), is the smallest number \(r\) such that there exists a non-dyadic partition \(\omega:=i_1,\dots, i_r\) of \(n\) with Stiefel-Whitney number \(\langle s_\omega(F),[F]\rangle \neq 0\). Here non-dyadic means that none of the \(i_t\) is of the form \(2^s-1\) and \(s_\omega(F)\) is the polynomial \(s_\omega(w_1,\dots, w_n)\in H^n(F;\mathbb{Z}_2)\) in the Stiefel-Whitney classes \(w_j=w_j(F)\) of \(F\) obtained from the monomial symmetric polynomial \(\sum x_1^{i_1}\dots x_r^{i_r}\) in the indeterminates \(x_1,\dots, x_n\) expressed as a polynomial \(s_\omega(\sigma_1,\dots, \sigma_n)\) in the elementary symmetric polynomials \(\sigma_j=\sigma_j(x_1,\dots, x_n)\).

The main result of the paper is the following: Let \(T:M\to M\) be a smooth involution of a closed \(m\)-dimensional manifold which is non-bounding and let \(F=Fix(T)\). Suppose that there exist integers \(2\leq j<n<m\) such that \(F^k=\emptyset\) if \(j<k<n\) and that the \(F^j\) is non-bounding. Then \(m\leq \mathcal{M}(n-j)+\ell(F^j)\). The authors provide examples to show, in some ‘special situations’ that the bound for \(m\) is best possible. Taking \(j=n-1\), the authors obtain an improvement on the five-halves theorem.

Reviewer: Parameswaran Sankaran (Trieste)