GYSIN TRIANGLES IN THE CATEGORY OF MOTIFS WITH MODULUS

In this article, we study a Gysin triangle in the category of motives with modulus (Theorem 1.2). We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives (Corollary 1.6). The corollary implies that an object in $\operatorname {\mathbf {MDM}^{\operatorname {eff}}}$ decomposes into a p-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.

Bounding size of homotopy groups of Spheres

Let p be prime. We prove that, for n odd, the p-torsion part of πq(Sn) has cardinality at most $p^{2^{{1}/({p-1})(q-n+3-2p)}}$ and hence has rank at most 21/(p−1)(q−n+3−2p). for p = 2, these results also hold for n even. The best bounds proven in the existing literature are $p^{2^{q-n+1}}$ and 2q−n+1, respectively, both due to Hans–Werner Henn. The main point of our result is therefore that the bound grows more slowly for larger primes. As a corollary of work of Henn, we obtain a similar result for the homotopy groups of a broader class of spaces.

An extension of properties of symmetric group to monoids and a pretorsion theory in a category of mappings

Several elementary properties of the symmetric group [Formula: see text] extend in a nice way to the full transformation monoid [Formula: see text] of all maps of the set [Formula: see text] into itself. The group [Formula: see text] turns out to be the torsion part of the monoid [Formula: see text]. That is, there is a pretorsion theory in the category of all maps [Formula: see text], [Formula: see text] an arbitrary finite set, in which bijections are exactly the torsion objects.

Two-torsion in the grope and solvable filtrations of knots

We study knots of order [Formula: see text] in the grope filtration [Formula: see text] and the solvable filtration [Formula: see text] of the knot concordance group. We show that, for any integer [Formula: see text], there are knots generating a [Formula: see text] subgroup of [Formula: see text]. Considering the solvable filtration, our knots generate a [Formula: see text] subgroup of [Formula: see text] [Formula: see text] distinct from the subgroup generated by the previously known [Formula: see text]-torsion knots of Cochran, Harvey, and Leidy. We also present a result on the [Formula: see text]-torsion part in the Cochran, Harvey, and Leidy’s primary decomposition of the solvable filtration.

Abelian groups whose nonzero endomorphisms have nonessential kernels

In this paper, we describe completely the [Formula: see text]-singular subgroup of an abelian group and a [Formula: see text]-nonsingular abelian group in terms of the basic subgroups of its [Formula: see text]-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a [Formula: see text]-nonsingular abelian group are [Formula: see text]-nonsingular and give a condition under which a pure extension of a [Formula: see text]-nonsingular abelian group by a [Formula: see text]-nonsingular group is [Formula: see text]-nonsingular.