[Paper Review] Notes on explicit smooth maps on 7-dimensional manifolds into the 4-dimensional Euclidean space

Naoki Kitazawa|arXiv (Cornell University)|Nov 25, 2019

Homotopy and Cohomology in Algebraic Topology21 references11 citations

TL;DR

This paper constructs explicit fold maps from 7-dimensional closed manifolds to 4-dimensional Euclidean space, analyzing their topological and geometric properties via Reeb spaces and cohomology rings. It establishes that certain 7-manifolds, including exotic spheres, do not admit special generic maps into R⁴, while providing new algebraic-topological invariants through the structure of fold map singularities and their induced maps on cohomology and homology groups.

ABSTRACT

A fold map is a smooth map at each singular point of which it is represented as the product map of a Morse function and the identity map on an open ball. A special generic map is a fold map such that the Morse function can be taken as a natural height function on an unit disk. The class of special generic maps includes a Morse function with exactly two singular points on a closed manifold, characterizing a sphere topologically (except 4-dimensional cases) as the Reeb's theorem shows, and canonical projections of unit spheres. It has been known that so-called exotic spheres do not admit special generic maps into Euclidean spaces whose dimensions are sufficiently high and smaller than the dimensions of the spheres. Exotic 7-dimensional homotopy spheres do not admit special generic maps into the 4-dimensional Euclidean space for example. We can easily obtain special generic maps on fundamental manifolds such as ones represented as connected sums of products of two standard spheres and in considerable cases, smooth manifolds resembling topologically them and different from them do not admit special generic maps. These interesting results are due to studies of Saeki, Sakuma and Wrazidlo since the 1990s. In the present paper, we present new results on explicit smooth maps including fold maps on 7-dimensional manifolds into the 4-dimensional Euclidean space and meanings in algebraic topology and differential topology of manifolds. Moreover, the author obtained related results before motivated by the studies before and they are reviewed in the presentation of the new results. We also present new discussions and results related to the results for 7-dimensional manifolds and maps on them for fold maps between manifolds of general dimensions.

Motivation & Objective

• To construct explicit smooth fold maps from 7-dimensional closed manifolds to 4-dimensional Euclidean space.
• To investigate the topological obstructions preventing exotic 7-dimensional homotopy spheres from admitting special generic maps into R⁴.
• To analyze the algebraic topology of manifolds admitting such fold maps, particularly focusing on cohomology rings and homology groups.
• To extend results on Reeb spaces of fold maps to cases where domain dimensions are not sufficiently high relative to the target.
• To provide a systematic study of the induced maps on cohomology and homology arising from fold maps with embedded singular sets and standard sphere fibers.

Proposed method

• Constructs fold maps between manifolds of general dimensions using local models with indices 0 and 1, based on the standard fold map form: $(x_1,\dots,x_m) \mapsto (x_1,\dots,x_{n-1}, \sum_{k=n}^{m-i} x_k^2 - \sum_{k=m-i+1}^{m} x_k^2)$.
• Applies the Reeb space construction to fold maps whose regular fibers are disjoint unions of standard spheres and whose singular set restrictions are embeddings.
• Analyzes the induced homomorphisms on cohomology and homology groups via pullbacks of characteristic classes and cup products.
• Uses the structure of the Reeb space to relate the cohomology ring of the domain manifold to the image of the fold map and the singular set.
• Applies known results from singularity theory and Morse theory to extend the classification of fold maps beyond the special generic case.
• Employs algebraic tools such as cup product relations and Poincaré duality to derive constraints on the cohomology rings of 7-manifolds admitting such maps.

Experimental results

Research questions

• RQ1 Which 7-dimensional manifolds admit fold maps into R⁴, and what are the topological obstructions to such maps?
• RQ2 Why do exotic 7-dimensional homotopy spheres not admit special generic maps into R⁴, and how does this relate to their smooth structure?
• RQ3 How do the cohomology rings and homology groups of 7-manifolds relate to the structure of their Reeb spaces under fold maps?
• RQ4 What algebraic invariants are preserved or induced by fold maps with embedded singular sets and standard sphere fibers?
• RQ5 To what extent can the Reeb space construction recover the cohomology ring structure of the domain manifold, especially when the domain dimension is not sufficiently high?

Key findings

• Exotic 7-dimensional homotopy spheres do not admit special generic maps into R⁴, confirming a known obstruction in differential topology.
• For fold maps $ f: M^7 \to \mathbb{R}^4 $ with embedded singular set and fibers diffeomorphic to $ S^3 $ or $ S^3 \sqcup S^3 $, the Reeb space inherits significant topological invariants.
• The cohomology ring of the 7-manifold $ M $ is constrained by the cup product structure of pullbacks from the Reeb space, particularly through relations like $ a_j^* \cdot \phi_{A,m-k}(a_j) $ generating $ H^m(M;\mathbb{Z}) $.
• Certain cup products vanish (e.g., $ a_j^* \cdot \phi_{n,m-n}((0,c)) = 0 $), indicating non-trivial relations in the cohomology ring induced by the fold map.
• The product $ b_{j,n}^* \cdot \phi_{n,m-n}((b_j,0)) $ forms a generator of $ H^m(M;\mathbb{Z}) $, showing that specific cohomology classes are linked to the singular set structure.
• The map $ f|_{S(f)} $ is an embedding, and all singular points have index 0 or 1, which ensures the Reeb space is well-behaved and allows for algebraic control of the topology.