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Register a new userOn the simplicial volume and the Euler characteristic of (aspherical) manifolds
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A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study vario
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We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth. Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case). However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4.
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.
Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). We consider the smallest infinity category $text{Top}^text{rx}$ containing both these classes of spaces and closed under homotopy pushout squares. In our main result, we compute the K-theory $K_0(text{Top}^text{rx})$, which is freely generated by equivalence classes of connected p-finite spaces, as p ranges over all primes. This provides a negative answer to Baezs question globally, but a positive answer when we restrict attention to a prime.
Building on work of Stolz, we prove for integers $0 le d le 3$ and $k>232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal-Williams conjecture in terms of the vanishing of a certain Toda bracket, and then to analyze this Toda bracket by bounding its $mathrm{H}mathbb{F}_p$-Adams filtrations for all primes $p$. We additionally prove new vanishing lines in the $mathrm{H}mathbb{F}_p$-Adams spectral sequences of spheres and Moore spectra, which are likely to be of independent interest. Several of these vanishing lines rely on an Appendix by Robert Burklund, which answers a question of Mathew about vanishing curves in $mathrm{BP} langle n rangle$-based Adams spectral sequences.
In ~cite{Iw2} Iwase has constructed two 16-dimensional manifolds $M_2$ and $M_3$ with LS-category 3 which are counter-examples to Ganeas conjecture: ${rm cat_{LS}} (Mtimes S^n)={rm cat_{LS}} M+1$. We show that the manifold $M_3$ is a counter-example to the logarithmic law for the LS-category of the square of a manifold: ${rm cat_{LS}}(Mtimes M)=2{rm cat_{LS}} M$. Also, we construct a map of degree one $$f:Nto M_2times M_3$$ which reduces Rudyaks conjecture to the question whether ${rm cat_{LS}}(M_2times M_3)ge 5$. We show that ${rm cat_{LS}}(M_2times M_3)ge 4$.
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