[Paper Review] Non-zero degree maps between $2n$-manifolds

Haibao Duan, Shicheng Wang|ArXiv.org|Feb 8, 2004

Geometric and Algebraic Topology10 references21 citations

TL;DR

This paper provides a computable necessary and sufficient condition for the existence of non-zero degree maps between (n−1)-connected 2n-manifolds using Thom-Pontrjagin duality and intersection forms. The key result is that for 4-manifolds, a degree-k map exists from a closed oriented 4-manifold M to a simply connected 4-manifold L if and only if the intersection form of L is isomorphic to a direct summand of that of M when k=1, generalizing to a matrix condition for arbitrary k.

ABSTRACT

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $ϕ: H^n(L;Z) o H^n(M;Z)$ can be realized by a map $f:M o L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds $M$ and $L$, $n>1$. A corollary is that each $(n-1)$-connected $2n$-manifold admits selfmaps of degree larger than 1, $n>1$. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree $k$ map from a closed orientable 4-manifold $M$ to a closed simply connected 4-manifold $L$ in terms of their intersection forms, in particular there is a map $f:M o L$ of degree 1 if and only if the intersection form of $L$ is isomorphic to a direct summand of that of $M$.

Motivation & Objective

• To determine when a homomorphism between cohomology groups of (n−1)-connected 2n-manifolds can be realized by a map of non-zero degree.
• To provide a necessary and sufficient condition for the existence of degree-k maps between closed, oriented (n−1)-connected 2n-manifolds with n>1.
• To extend the characterization to 4-manifolds using intersection forms, particularly for maps to simply connected 4-manifolds.
• To show that every (n−1)-connected 2n-manifold admits self-maps of degree greater than 1 for n>1.
• To establish a complete homotopical and algebraic classification of such maps via matrix equations and homotopy groups.

Proposed method

• Uses Thom-Pontrjagin duality to relate maps between manifolds to homomorphisms on cohomology and homology, particularly focusing on the free parts of H^n.
• Applies the intersection form X_M defined on the free part of H^n(M), represented as a unimodular matrix A under a chosen basis.
• Derives a matrix condition P^T A P = kB, where P encodes the induced map on homology, and B is the intersection form of the target manifold.
• Incorporates secondary homotopy invariants via Whitehead products [s^n, s^n] in π_{2n−1}(S^n) to account for the non-trivial attaching maps of the 2n-cell.
• For 4-manifolds, proves that the existence of a degree-k map f:M→L is equivalent to the matrix equation P^T A P = kB, with additional constraints on the homotopy class of f.
• Uses surgery and classification results to show that the number of homotopy classes of such maps is finite and controlled by π_{2n}(L).

Experimental results

Research questions

• RQ1 When does a given homomorphism φ:H^n(L;Z)→H^n(M;Z) arise from a map f:M→L of non-zero degree k?
• RQ2 What is the precise algebraic condition on the intersection forms and homotopy invariants that guarantees the existence of a degree-k map between (n−1)-connected 2n-manifolds?
• RQ3 For 4-manifolds, when does a degree-1 map exist from M to L, and how is this related to the structure of their intersection forms?
• RQ4 How many homotopy classes of maps realize a given degree-k map between such manifolds?
• RQ5 Can every (n−1)-connected 2n-manifold admit self-maps of degree greater than 1 for n>1?

Key findings

• A degree-k map f:M→L exists between (n−1)-connected 2n-manifolds if and only if the matrix P satisfies P^T A P = kB and a secondary homotopy condition involving Whitehead products.
• For 4-manifolds, a degree-k map f:M→L exists if and only if P^T A P = kB, where A and B are the intersection forms of M and L under chosen bases.
• A degree-1 map f:M→L exists if and only if the intersection form of L is isomorphic to a direct summand of that of M.
• Every (n−1)-connected 2n-manifold admits self-maps of degree greater than 1 for n>1, as a corollary of the main theorem.
• The number of homotopy classes of maps realizing a given degree-k map is finite and in bijection with elements of π_{2n}(L).
• For 4-manifolds, D(M,L) = {0} in cases like CP²#CP² and S²×S², while D(T⁴, #₃S²×S²) = Z, showing that every integer degree is realizable.