On distributional one-category, diagonal distributional complexity, and related invariants

Homotopy type theory is a version of Martin-Lof type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RPn, as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RPn by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP-1 to be the empty type. In the inductive step, we take RPn+1 to be the mapping cone of the projection map of the tautological bundle of RPn, and we use its universal property and the univalence axiom to define the tautological bundle on RPn+1.By showing that the total space of the tautological bundle of RPn is the n-sphere Sn, we retrieve the classical description of RPn+1 as RPn with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP∞, defined as the sequential colimit of RPn with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles.These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.

For a simplicial augmented algebra K, Eilenberg–Mac Lane constructed a chain map . They proved that g is a reduction (homology isomorphism) and conjectured that it is also the injection of a contraction (special homotopy equivalence). The contraction is followed at once by using homological perturbation techniques. If K is commutative, Eilenberg–Mac Lane proved that g is a morphism of DGA-algebras. The present article is devoted to proving that f and φ satisfy certain multiplicative properties (weaker than g) and showing how they can be used for computing in an economical way the homology of twisted cartesian products of two Eilenberg–Mac Lane spaces.

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and \operatorname{CAT}(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete \mathcal{VH} complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. For the class of complete directed \mathcal{VH} complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise. Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete \mathcal{VH} complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F_2 .

Let T(j) be the dual of the j th stable summand of Ω2 S 3 (at the prime 2) with top class in dimension j. Then it is known that T(j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T(j)→ T(2j)→… is an infinite wedge of stable summands of K(V,1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T(1), one gets K(Z/2, 1) = RP ∞ as one of the summands.I discuss a generalization of this picture using higher iterated loopspaces and Eilenberg MacLane spaces. I consider certain finite spectra T(n, j) for n, j ≥ 0 (with T(1, j) = T(j)), dual to summands of Ω n+1 S N , conjecture generalizations of the above, and prove that these conjectures are correct in cohomology. So, for example, T(n,j) has unstable cohomology, and the cohomology of the hocolimit of a certain sequence T(n,j) → T(n, 2j) → …. agrees with the cohomology of the wedge of stable summands of K(V,n)’s corresponding to the wedge occurring in the n = 1 case above.One can also map the T(n, j) to each other as n varies, and here the cohomological calculations imply a homotopical conclusion: the hocolimits that are nonzero, T(∞, 2k), for k ≥ 0, map to each other, giving rise to a fitration of H Z/2 which is equivalent to the mod 2 symmetric powers of spheres filtration.Our homotopical constructions use Hopf invariant methods and loop-space technology. These are quite general and should be of independent interest. To study the action of the Steenrod operations on the cohomology of our spectra, we derive a Nishida formula for how X(Sq i) acts on Dyer—Lashof operations. This should be of use in other settings. In an appendix, we explain connections with recent work by Greg Arone and Mark Mahowald on the Goodwillie tower of the identity.

Let $X$ be a finite CW-complex with finite fundamental group. We show that the group ${\text {aut}}(X)$ of homotopy classes of self-homotopy equivalences of $X$ is commensurable to an arithmetic group. If in addition $X$ is an oriented manifold then the subgroup ${\text {au}}{{\text {t}}_t}(X)$ of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if $X$ is a smooth manifold of dimension $\geqslant 5$ then the subgroup ${\text {diff}}(X)$ of ${\text {aut}}(X)$ the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.

The purpose of this paper is to study extendibility and stable extendibility of vector bundles over real projective spaces. We determine a necessary and sufficient condition that a vector bundle ζ over the real projective n-space RPn is extendible (or stably extendible) to RPm for every m>n in the case where ζ is the complexification of the tangent bundle of RPn and in the case where ζ is the normal bundle associated to an immersion of RPn in the Euclidean (n+k)-space Rn+k or its complexification, and give examples of the normal bundle which is extendible to RPN but is not stably extendible to RPN+1.

The s-th higher topological complexity TC s ⁡ ( X ) {\operatorname{TC}_{s}(X)} of a space X can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when X = ℝ ⁢ P m {X=\operatorname{\mathbb{R}P}^{m}} , the real projective space of dimension m. In particular, we describe a number r ⁢ ( m ) {r(m)} , which depends on the structure of zeros and ones in the binary expansion of m, and with the property that 0 ≤ s ⁢ m - TC s ⁡ ( ℝ ⁢ P m ) ≤ δ s ⁢ ( m ) {0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)} for s ≥ r ⁢ ( m ) {s\geq r(m)} , where δ s ⁢ ( m ) = ( 0 , 1 , 0 ) {\delta_{s}(m)=(0,1,0)} for m ≡ ( 0 , 1 , 2 ) mod 4 {m\equiv(0,1,2)\bmod 4} . Such an estimation for TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} appears to be closely related to the determination of the Euclidean immersion dimension of ℝ ⁢ P m {\operatorname{\mathbb{R}P}^{m}} . We illustrate the phenomenon in the case m = 3 ⋅ 2 a {m=3\cdot 2^{a}} . In addition, we show that, for large enough s and even m, TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} is characterized as the smallest positive integer t = t ⁢ ( m , s ) {t=t(m,s)} for which there is a suitable equivariant map from Davis’ projective product space P 𝐦 s {\mathrm{P}_{\mathbf{m}_{s}}} to the ( t + 1 ) {(t+1)} -st join-power ( ( ℤ 2 ) s - 1 ) ∗ ( t + 1 ) {((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}} . This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating TC 2 {\operatorname{TC}_{2}} to the immersion dimension of real projective spaces.

In this paper, actions of finite groups on aspherical spaces are studied. A complete homotopy classification of free actions of finite groups on aspherical spaces is obtained, including some results in nonfree actions. For example, a relation between cohomologies of finite groups and the lattice structure of their subgroups is obtained by this method; this relation is formulated in terms of spectral sequences.

Isotopies of 3-manifolds

We study the structure of the formal groups associated to the Morava $K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$ for a fixed $n$ enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonn\'e correspondence between bicommutative Hopf algebras and Dieudonn\'e modules. We extend P. Goerss's results on the bilinear products of such Hopf algebras and corresponding Dieudonn\'e modules.

Analysis and Algebra on Differentiable Manifolds

Let P r {P_r} be r-dimensional real projective space with r odd, and let π 0 Diff + : P r {\pi _0}{\text {Diff}^ + }:{P_r} be the group of orientation preserving diffeomorphisms P r → P r {P_r} \to {P_r} factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for r ≡ 11 mod 16 r \equiv 11 \bmod 16 the group π 0 Diff + : P r {\pi _0}{\text {Diff}^ + }:{P_r} is isomorphic to the homotopy group π r + 1 + k ( P ∞ / P k − 1 ) {\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}}) , where k = d 2 L − r − 1 k = d{2^L} - r - 1 with L ≥ φ ( ( r + 1 ) / 2 ) L \geq \varphi ((r + 1)/2) and d 2 L ≥ r + 1 d{2^L} \geq r + 1 . The function φ \varphi is denned by φ ( l ) = { i | 0 > i ≤ l , i ≡ 0 , 1 , 2 , 4 mod ( 8 ) } \varphi (l) = \{ i|0 > i \leq l,i \equiv 0,1,2,4 \bmod (8)\} . The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism t : π 0 Diff + : P r → Ω r + 1 ( v ) t:{\pi _0}{\text {Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v) for r ≡ 11 mod 16 r \equiv 11 \bmod 16 and Ω r + 1 ( v ) {\Omega _{r + 1}}(v) a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup Ω r + 1 ( v ) {\Omega _{r + 1}}(v) , and that this subgroup is isomorphic to π r + 1 + k ( P ∞ / P k − 1 ) {\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}}) as above. Then one reads off from Mahowald’s tables of π n + m ( P ∞ / P m − 1 ) {\pi _{n + m}}({P_\infty }/{P_{m - 1}}) that π 0 Diff + : P 11 = Z 2 {\pi _0}{\text {Diff}^ + }:{P_{11}} = {Z_2} and π 0 Diff + : P 27 = 6 Z 2 {\pi _0}{\text {Diff}^ + }:{P_{27}} = 6{Z_2} .

We give a complete classification of the Dehn surgeries on Montesinos knots which yield manifolds with cyclic or finite fundamental groups.

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension ≥ 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension ≥ 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed oriented manifolds of dimension ≥ 5 with non-spin universal cover, with finite fundamental group having all Sylow subgroups elementary abelian, admit metrics of positive scalar curvature, once one restricts to the complement of manifolds whose homology classes are toral. The exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.

Linkings and Pfaff's equations D Bennequin The topology of four-dimensional manifolds Yu P Solov'ev Singularities of systems of rays Vladimir I Arnol'd On the geometry of hyperbolic attractors of smooth cascades Romen V Plykin Surgery on manifolds with finite fundamental groups A F Kharshiladze Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Cell-like maps. The problem of raising dimension A N Dranishnikov and E V Shchepin IMMERSIONS AND EMBEDDINGS OF SMOOTH MANIFOLDS