🔢 mathematics

Invariants of real affine varieties based on their complexifications

This paper introduces a new family of invariants for real algebraic sets based on their complexifications, which are used to classify algebraic vector bundles over products of spheres, determine the existence of regular maps between spheres, and disprove a conjecture regarding weak algebraic approximation.

Original authors: Juliusz Banecki

Published 2026-05-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juliusz Banecki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a shape made of clay (a smooth surface) and you want to know if you can recreate that exact shape using only rigid, mathematical building blocks (polynomial equations). This is the heart of the problem Juliusz Banecki tackles in his paper.

Here is a simple breakdown of what he did, using everyday analogies.

The Main Problem: The "Rigid vs. Flexible" Puzzle

In the world of math, there are smooth shapes (like a perfect sphere or a twisted knot) that are flexible. You can stretch or squish them, and they are still considered the "same" shape. Then there are algebraic shapes, which are defined by strict rules (equations). These are like sculptures made of steel; they are rigid and can't be bent without breaking the rules.

Mathematicians have long known that for almost any smooth shape, you can find a rigid algebraic version that looks exactly like it. This is like saying, "For every clay sculpture, there is a steel version that looks identical."

However, a harder question remains: If you have two rigid shapes, can you draw a path from one to the other using only rigid lines?

Imagine trying to walk from a steel sphere to a steel donut. Can you do it without ever stepping off the "rigid" path?
Sometimes the answer is "No," even if a flexible path exists. The rigid rules block the way.

The New Tool: The "Complex Mirror"

To figure out when these rigid paths are blocked, Banecki invented a new set of tools called invariants. Think of an invariant as a "fingerprint" or a "security code" for a shape.

His clever trick is to look at the shape through a complex mirror.

The Real Shape: This is the object we see in our normal world (like a sphere in 3D space).
The Complexification: This is a "shadow" or a "reflection" of the real shape in a higher-dimensional, more magical world (complex numbers).

Banecki's insight is that the "fingerprint" of the real shape is hidden inside the structure of this complex mirror. By studying the complex mirror, he can calculate a code that tells him if a rigid path is possible. If the codes of two shapes don't match up in a specific way, you know for sure that no rigid path exists between them.

The Big Discoveries

1. The "Sphere Product" Mystery

The paper focuses heavily on shapes made by multiplying two spheres together (like a donut, which is a circle times a circle, but in higher dimensions).

The Old Rule: Mathematicians knew that if you tried to map a product of two spheres into a bigger sphere, sometimes it was impossible.
The New Discovery: Banecki found a specific "forbidden zone." He proved that if you take two spheres of certain sizes (specifically where the numbers behave in a certain way related to the number 4), you cannot create a rigid map between them that has an "odd" number of twists.
The Metaphor: Imagine trying to tie a knot in a rope. If the rope is made of steel, you can only tie knots in a specific way. Banecki proved that for certain rope lengths, you can never tie a "left-handed" knot, only "right-handed" ones (or vice versa). This disproved a long-held guess that you could always tie any knot you wanted.

2. The "Weak Approximation" Trap

There is a concept called "weak algebraic approximation." Imagine you have a smooth, wobbly jelly shape. You want to replace it with a rigid steel shape that is almost the same, but maybe has a tiny crack or a rough spot (the "nonsingular locus").

The Old Belief: Mathematicians thought that if the "jelly" and the "steel" were close enough in a basic sense (like having the same number of holes), you could always find a steel version, even if it had a rough spot.
The New Discovery: Banecki proved this is false. He found a specific jelly shape and a specific steel container where, even though they look similar and fit the basic rules, the jelly cannot be turned into a steel shape, even with a rough spot.
The Metaphor: It's like trying to fit a soft, squishy pillow into a rigid box. You might think, "If I just squish it a little, it will fit." Banecki showed a case where, no matter how much you squish it, the pillow simply refuses to fit into the box's rigid geometry, even if you allow the box to be slightly dented.

Why This Matters

Before this paper, mathematicians had to guess whether certain rigid paths existed or if certain shapes could be approximated by rigid ones. They often relied on tools that were too weak to see the problem.

Banecki's new "complex mirror" tools are like putting on X-ray glasses. They allow mathematicians to see the hidden structural barriers that prevent rigid shapes from connecting or approximating each other. He didn't just find a few new examples; he completely solved the classification of these rigid paths for products of spheres and proved that a major mathematical guess was wrong.

In short: He built a new mathematical detector that tells us exactly when the rigid rules of algebra make a shape impossible to reach, solving puzzles that had stumped experts for decades.

Title: Invariants of Real Affine Varieties Based on Their Complexifications
Author: Juliusz Banecki

1. Problem Statement

The paper addresses fundamental realization problems in real algebraic geometry, specifically concerning the relationship between topological data and algebraic structures. The core issues investigated are:

Algebraic Realization of Homotopy Classes: Determining which homotopy classes of continuous maps between real algebraic sets (specifically products of spheres) can be represented by regular (algebraic) maps.
Algebraic Approximation: Investigating whether smooth submanifolds of real algebraic sets admit "weak algebraic approximations" (approximations by the nonsingular locus of a Zariski closed subset).
Limitations of Existing Invariants: Current invariants based on singular homology/cohomology or standard algebraic K-theory are often too weak to obstruct the existence of regular maps in certain cases (e.g., between spheres of different dimensions) or to distinguish between algebraic and non-algebraic approximation scenarios.

2. Methodology

The author introduces a new family of invariants defined via the topology of the complexifications of real affine varieties. The methodology relies on the following steps:

Complexification: For a nonsingular complete real affine variety $X$ , one considers a nonsingular projective complexification $\mathcal{X}$ (a complex variety defined over $\mathbb{R}$ such that its real locus $X$ is Zariski dense in $\mathcal{X}$ ).
Generalized Cohomology Theories: Instead of using singular cohomology, the paper employs stronger generalized cohomology theories. Specifically, it utilizes:
- Complex-oriented theories ( $E^*$ ): Such as complex K-theory ( $KU^*$ ) and complex cobordism.
- Real-oriented equivariant theories ( $F^\star$ ): Such as Atiyah's Real K-theory ( $KR^\star$ ) and Real cobordism, which account for the $\mathbb{Z}/2$ -action induced by complex conjugation on the complexification.
Definition of Invariants: For a theory $E^*$ (or $F^\star$ ), the invariant $E^*_C(X)$ (or $F^\star_C(X)$ ) is defined as the image of the cohomology of the complexification under the inclusion of the real locus:
$E^*_C(X) := \text{im}(i^*: E^*(\mathcal{X}) \to E^*(X))$
The paper proves that this image is independent of the choice of the complexification $\mathcal{X}$ .
Comparison with Algebraic K-Theory: The author establishes that these new invariants contain the classical algebraic K-groups (e.g., $K_0(R(X, \mathbb{C}))$ ) as subgroups, thereby providing a computable framework that is strictly stronger than singular homology invariants but more tractable than raw algebraic K-theory.

3. Key Contributions and Results

A. Computation of Invariants for Spheres

The paper provides a complete computation of the invariants $KU^*_C(S^n)$ and $KR^\star_C(S^n)$ for all $n$ .

Unitary Case: $KU^p_C(S^n) = KU^p(S^n)$ if $n$ is even, and $0$ if $n$ is odd.
Orthogonal/Real Case: The structure of $KO^p_C(S^n)$ is determined by the congruence of $n$ modulo 4 and 8. The index of the subgroup $KO^p_C(S^n)$ within the full topological group $KO^p(S^n)$ is described by a function $\phi(n, m)$ dependent on the binary expansion of the dimensions.

B. Classification of Algebraic Vector Bundles

Using the computed invariants, the paper completely classifies the topological isomorphism classes of algebraic vector bundles over products of two spheres $S^n \times S^m$ .

Theorem: The image of the algebraic K-group $\tilde{K}_0(R(S^n \times S^m, \mathbb{K}))$ (where $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$ ) in the corresponding topological K-group is explicitly described as a direct sum involving subgroups of index determined by $\phi(n, m)$ .
This resolves the long-standing problem of computing $K_0(R(S^2 \times S^2, \mathbb{H}))$ and generalizes previous results by Bochnak and Kucharz.

C. Regular Maps Between Spheres

The invariants are applied to determine the existence of regular maps $f: S^n \times S^m \to S^{n+m}$ .

Disproof of a Conjecture: The paper disproves Conjecture 1.11 (Bochnak and Kucharz, 2022), which posited that every continuous map from a nonsingular compact real algebraic set of dimension $n$ $n$ to $S^n$ $S^{n}$ is homotopic to a regular one.
- Result: If $n \equiv 1 \pmod 4$ and $n > 1$ , there exist continuous maps $S^n \times S^m \to S^{n+m}$ that are not homotopic to any regular map.
New Obstructions: The paper identifies specific conditions (e.g., one dimension $\equiv 2 \pmod 4$ and the other $\equiv 2, 3 \pmod 4$ ) where a regular map exists if and only if its topological degree is even.
Positive Results: The author generalizes Loday's construction using "nice normed bilinear maps" to show that if $n \le 8$ and the binomial coefficient $\binom{n+m}{n}$ is odd, every continuous map $S^n \times S^m \to S^{n+m}$ is homotopic to a regular one. This supports a new conjecture (Conjecture 1.15) linking the existence of regular maps to the parity of $\binom{n+m}{n}$ .

D. Obstructions to Algebraic Approximation

The paper addresses the problem of weak algebraic approximation of submanifolds.

Disproof of a Conjecture: It disproves a conjecture from [26] suggesting that the algebraicity of the bordism class of the inclusion $[M \hookrightarrow X]$ is sufficient for the existence of a weak algebraic approximation.
Counterexample: The author constructs a nonsingular compact real algebraic set $X$ $X$ of dimension 6 and a smooth submanifold $M$ $M$ of dimension 4 such that:
1. The bordism class of the inclusion is zero (algebraic).
2. $M$ does not admit a weak algebraic approximation in $X$ .
Mechanism: The obstruction is provided by the new invariants (specifically using $H^*_C(X; \mathbb{Z})$ and Real-oriented theories), which detect that the required cohomology classes cannot be realized algebraically, even when the bordism obstruction vanishes.

4. Significance

The paper claims significance in three main areas:

Theoretical Advancement: It introduces a robust, computable family of invariants ( $E^*_C$ and $F^\star_C$ ) that bridge the gap between elementary singular homology invariants and the difficult-to-compute algebraic K-groups. These invariants are shown to be independent of the choice of complexification.
Resolution of Open Problems: It provides the first complete classification of algebraic vector bundles over products of spheres and settles the status of Conjecture 1.11 regarding regular maps from algebraic sets to spheres.
Refutation of Sufficiency Conditions: It demonstrates that the algebraicity of the bordism class of an inclusion is not sufficient for weak algebraic approximation, thereby refining the understanding of the obstructions in real algebraic geometry. The results suggest that stronger invariants based on Real-oriented cohomology are necessary to fully characterize these approximation problems.

The paper concludes by noting that while the current invariants (based on K-theory) are sufficient for the presented counterexamples, even stronger theories (like Real cobordism) might be required to resolve remaining open cases, such as Conjecture B(p) in [26].