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Register a new userStabilization of the homotopy groups of the self equivalences of linear spheres
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Authors
Assaf Libman
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Let $G$ be a finite group. Let $U_1,U_2,dots$ be a sequence of orthogonal representations in which any irreducible representation of $oplus_{n geq 1} U_n$ has infinite multiplicity. Let $V_n=oplus_{i=1}^n U_n$ and $S(V_n)$ denote the linear sphere of unit vectors. Then for any $i geq 0$ the sequence of group $dots rightarrow pi_i operatorname{map}^G(S(V_n),S(V_n)) rightarrow pi_i operatorname{map}^G(S(V_{n+1}),S(V_{n+1})) rightarrow dots$ stabilizes with the stable group $oplus_H omega_i(BW_GH)$ where $H$ runs through representatives of the conjugacy classes of all the isotropy group of the points of $S(oplus_n U_n)$.
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Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $aut_G(S(V)^*)$ denote the space of $G$-equivariant self homotopy equivalences of $S(V)^{*n}$. We show that for any $k geq 1$ there exists $M>0$ which depends only on $V$ such that $|pi_k aut_G(S(V)^{*n})| leq M$ is for all $n gg 0$.
We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a new computational method that yields a streamlined computation of the first 61 stable homotopy groups, and gives new information about the stable homotopy groups in dimensions 62 through 90. The method relies more heavily on machine computations than previous methods, and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence.
These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007.
We prove that the 2-primary $pi_{61}$ is zero. As a consequence, the Kervaire invariant element $theta_5$ is contained in the strictly defined 4-fold Toda bracket $langle 2, theta_4, theta_4, 2rangle$. Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the only ones are $S^1, S^3, S^5$ and $S^{61}$. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.
Let $M$ be a topological monoid with homotopy group completion $Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $pi_k (Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of nullhomotopic maps. We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, leading to results about lifting continuous families of characters to continuous families of representations.
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