Quadratic functors and metastable homotopy

We investigate the rational homotopy classification problem for the components of some function spaces with Sn or cPn as target space.

/pdf/rational-homotopy-of-spaces-of-maps-into-spheres-and-complex-qdpplibxxd.pdf

Rational homotopy of spaces of maps into spheres and complex projective spaces

We explore the existence of homomorphisms between outer automorphism groups of free groups Out(F
 n
 ) → Out(F
 m
 ). We prove that if n > 8 is even and n ≠ m ≤ 2n, or n is odd and n ≠ m ≤ 2n − 2, then all such homomorphisms have finite image; in fact they factor through det : $${{\rm Out}(F_n) \to \mathbb{Z}/2}$$
 . In contrast, if m = r
 n
 (n − 1) + 1 with r coprime to (n − 1), then there exists an embedding $${{\rm Out}(F_n) \hookrightarrow {\rm Out}(F_m)}$$
 . In order to prove this last statement, we determine when the action of Out(F
 n
 ) by homotopy equivalences on a graph of genus n can be lifted to an action on a normal covering with abelian Galois group.

Abelian covers of graphs and maps between outer automorphism groups of free groups

Every closed oriented manifold $M$ is associated with a set of integers $D(M)$, the set of self-mapping degrees of $M$. In this paper we investigate whether a product $M\times N$ admits a self-map of degree $d$, when neither $D(M)$ nor $D(N)$ contains $d$. We find sufficient conditions so that $D(M\times N)$ contains exactly the products of the elements of $D(M)$ with the elements of $D(N)$. As a consequence, we obtain manifolds $M\times N$ that do not admit self-maps of degree $-1$ (strongly chiral), that have finite sets of self-mapping degrees (inflexible) and that do not admit any self-map of degree $dp$ for a prime number $p$. Furthermore we obtain a characterization of odd-dimensional strongly chiral hyperbolic manifolds in terms of self-mapping degrees of their products.

Degrees of Self-Maps of Products

We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y) We also discuss work on the homotopy theory of the monoid of self-equivalences aut(X) and of the free loop space LX We consider these topics in both ordinary homotopy theory as well as after localization In the latter case, we discuss algebraic models for the localization of function spaces and their applications

The homotopy theory of function spaces: a survey

The Homotopy of Map(Cpm, CPn)

On Degrees of Mapping

/pdf/highly-connected-poincare-complexes-1yfoecn7ha.pdf

Highly connected Poincaré complexes

/pdf/self-homotopy-equivalences-of-the-total-spaces-of-a-sphere-56itbusenu.pdf

Self-homotopy equivalences of the total spaces of a sphere bundle over a sphere

https://projecteuclid.org/download/pdf_1/euclid.kmj/1138036483

Seiya Sasao

Papers

The Homotopy of Map(Cpm, CPn)

On Degrees of Mapping

Highly connected Poincaré complexes

Self-homotopy equivalences of the total spaces of a sphere bundle over a sphere

On the group ε(K(π, 1)×X) for 1-connected CW-complexes X