Calculating higher homotopy groups of topological spaces is notoriously difficult. Even for something as simple as spheres—the fundamental building blocks of shape—we lack a complete formula for all higher homotopy groups.
Yet, amidst this wild complexity, there is a remarkable organizing principle: . Formally, the Freudenthal Suspension Theorem guarantees that the homotopy groups are eventually constant for , allowing us to define the stable homotopy groups of spheres via the direct limit:
In this blog post, we are going to prove this result, but not through standard algebraic machinery. Instead, we will use methods from Morse theory—drawing on the geometric framework in Milnor’s classic text—to build our proof from the ground up, introducing the necessary Morse-theoretic basics along the way.
In the second half of the post, we will shift to a completely different vantage point: higher homotopy groups through the lens of the . This geometric perspective translates abstract homotopy classes into framed cobordisms, giving us a deeply visual way to understand the stable homotopy groups of spheres.
To understand how Morse theory provides a powerful toolkit for topology, let us begin with a classic, intuitive example: the standard torus standing upright, equipped with the height function given by projection onto the vertical axis.
For any real number , let us define the sublevel set:
As we sweep the level parameter from the bottom of the torus to the top, the topology of the sublevel sets evolves. There are four critical points of the height function where the derivative vanishes, corresponding to the bottom-most minimum, the two saddle points on the sides, and the top-most maximum. Let us label these critical points:
Let’s examine how the topology of changes as varies across different intervals:
Building upon our intuitive geometric example of the upright torus, we can now formalize the foundational concepts and theorems of finite-dimensional Morse theory.
A smooth function on a smooth manifold is called a if all of its critical points (points where the differential ) are .
A critical point is nondegenerate if the Hessian matrix of second partial derivatives evaluated at , is non-singular (i.e., its determinant is non-zero, meaning it has no zero eigenvalues). Consequently, nondegenerate critical points are isolated.
If is a nondegenerate critical point of , the of at (often denoted ) is defined as: For our upright torus height function:Now, we know from our knowledge of differential topology that at points where we have immersions and submersions, it is essentially an inclusion and projection respectively by some coordinate change. An analogous theorem holds for nondegenerate critical points, which we call the Morse Lemma.
Morse’s Lemma : Let be a non-degenerate critical point for a smooth function . >Then there is a local coordinate system in a neighborhood of >with for all such that the identity holds throughout , where is the index of at .
Now we state the fundamental theorem of finite dimensional morse theory (there will be an analog for infinite dimensions later), which is really a generalization of what we had already seen in the example of the height function of the torus.
Fundamental Theorem: Let be a smooth function, and let be a >non-degenerate critical point with index . Setting , suppose that is compact, and contains no critical point of >other than , for some . Then, for all sufficiently small >, the set has the homotopy type of with a -cell attached.
As an illustration of the Fundamental Theorem, the picture below is the cell decomposition for the height function on the torus as it passes the nondegenerate critical points of index 1,1.
Now, having discussed Morse functions, one might intuitively think that such functions are rare or sparse. In reality, the exact opposite is true: we can obtain a plethora of Morse functions by slightly perturbing any smooth function, as guaranteed by the following result:
Lemma : Any bounded smooth function can be uniformly >approximated by a smooth function which has no degenerate critical points. Furthermore>, can be chosen so that the -th derivatives of on the compact set >uniformly approximate the corresponding derivatives of , for .
We can do even better: instead of arbitrary approximations, we can choose to be a Morse function that is sufficiently close to , ensuring that the index of critical points is preserved. This stability is guaranteed by the following lemma:
Let be a compact subset of the Euclidean space ; let be a neighborhood of ; and let be a smooth function such that all critical points of in have index .
Lemma : If is any smooth function which is “close” to , in >the sense that uniformly throughout , for some sufficiently small constant , then all >critical points of in have index .
Note that is allowed to have degenerate critical points. In the application, will be a nearby function without degenerate critical points. \end{quote}
So far, we have explored finite-dimensional Morse theory on compact manifolds like our upright torus. But why take this machinery further?
If our ultimate goal is to understand the of a manifold —which capture complex, multi-dimensional topological structures that the fundamental group alone cannot see—our objective is to establish the fundamental topological isomorphism relating a manifold to its loop space:
Theorem (Loop Space Isomorphism): For any path-connected manifold and base point >, there is a natural isomorphism of homotopy groups:
This identity reveals a profound shift in perspective: instead of tackling the difficult higher homotopy groups of a finite-dimensional manifold directly, we can study the homotopy groups of its infinite-dimensional loop space instead.To analyze the topology of , we need a version of Morse theory that can operate on infinite-dimensional spaces. This exact motivation leads us to adapt Morse theory for path and loop spaces, where geodesics will emerge as the critical points of the energy functional!
To rigorously connect these spaces, we rely on the machinery of fiber spaces.
A continuous map is called a if it satisfies the homotopy lifting property with respect to any space : for any homotopy and any map such that , there exists a lifted homotopy such that and .
are a weaker version of Hurewicz fibrations, requiring the homotopy lifting property to hold only for the class of CW-complexes (or cubes ) rather than all topological spaces.
Fibrations are uniquely valuable because they give us a powerful long exact sequence of homotopy groups:
Theorem (Long Exact Sequence of Homotopy Groups): Let be a Hurewicz >fibration with fiber over a base point . Then there is a >natural long exact sequence:
To prove our target isomorphism, we construct a canonical fibration using the path space and loop space of . Let denote the path space of (all continuous paths starting at the fixed base point ), and let be the loop space of (paths that both start and end at ).
Consider the evaluation map at the endpoint:
Applying the long exact sequence of homotopy groups to the fibration :
Since is contractible, all of its homotopy groups vanish for . By exactness of the sequence, the boundary homomorphism becomes an isomorphism for all .
Thus, shifting the index yields:
This exact sequence and fibration framework validate our infinite-dimensional strategy: analyzing the topology of through Morse theory directly unlocks the higher homotopy groups of , where will emerge as critical points of the energy functional!
Having established the algebraic topology foundations and the motivation for using infinite-dimensional spaces, we now transition to the framework of Morse theory on path spaces.
Let be a smooth manifold and let and be two (not necessarily distinct) points of [cite: 2]. By a from to will be meant a map such that:
The set of all piecewise smooth paths from to in will be denoted by , or briefly by or . Later (in Chapter 16) will be given the structure of a topological space, but for the moment this will not be necessary. We will think of as being something like an ``infinite dimensional manifold’’.
To start the analogy, by the will be meant the vector space consisting of all piecewise smooth vector fields along for which and . The notation will be used for this vector space.
Since each belongs to , note that , for all .
We will use either or to refer to the variation. More generally if, in the above definition, is replaced by a neighborhood of in , then (or ) is called an -parameter variation of . Now may be considered as a smooth path in . Its velocity vector is defined to be the vector field along given by: Clearly . This vector field is also called the associated with the variation .
Continuing with the analogy developed in the preceding section, we now wish to define a bilinear functional when is a critical point of the function , i.e., a geodesic. This bilinear functional will be called the .
To define , given vector fields choose a 2-parameter variation where is a neighborhood of in , so that
Then the Hessian will be defined to be the second partial derivative where denotes the path . This second derivative will be written briefly as . Defintion: The index of the Hessian is defined to be the maximum dimension of a subspace of on which is negative definite.
Just like the height function on the torus in finite-dimensional Morse theory, we now want to choose a special function on the path space to develop our infinite-dimensional Morse theory. Natural candidates include the length of a path or the energy of a path, which map .
However, to even define the length or energy of a path in , we need a way to measure distances and speeds. Therefore, henceforth we will assume that is a Riemannian manifold equipped with a Riemannian metric .
For any piecewise smooth path , we define the energy functional by:
Using the energy functional rather than arc length is a standard and analytically convenient choice in infinite-dimensional Morse theory because is smooth, whereas the length functional suffers from lack of differentiability at constant paths.
The primary motivation for studying the energy functional on path spaces is its deep connection to differential geometry. The critical paths of correspond exactly to the paths that satisfy the geodesic equations.
Theorem: A path is a critical point for the energy functional > if and only if is a smooth geodesic connecting >to .
Analyzing the Hessian and the index of a critical path requires advanced techniques and tools from Riemannian geometry. In particular, there is a precise characterization of the index of a path using Jacobi fields and conjugate points, which we will not get into here. However, a fundamental result flowing from this geometric framework is formally captured by the following theorem:
Theorem (Finiteness of the Morse Index) The Morse index of any critical >point (geodesic) of the energy functional on is always finite.
The space as a whole is infinite-dimensional. However, by restricting our attention to paths with bounded energy, we can construct finite-dimensional models that are also deformation retracts. Specifically, since a deformation retract preserves homotopy type, these finite-dimensional models share the exact same higher homotopy groups as the corresponding sublevel sets of .
Given , let denote the closed subset and let denote the open subset (where is the energy functional)[cite: 5]. We study the topology of by constructing a finite-dimensional approximation to it.
Choose a subdivision of the unit interval. Let be the subspace of consisting of paths such that:Finally, we define the restricted subspaces:
Lemma : Let be a complete Riemannian manifold, and let be a fixed positive number such that . Then for all sufficiently fine subdivisions of , the set can be given the structure of a smooth finite-dimensional manifold in a natural way.
To shorten notation, let us denote this manifold of broken geodesics by . Let denote the restriction to of the energy function .
Theorem : This function is smooth. Furthermore, for each , the set is compact, and is a of the corresponding set . The critical points of are precisely the same as the critical points of in : namely the unbroken geodesicsfrom to of length less than . The index (or the nullity) of the Hessian > at each such critical point is equal to the index (or the nullity) of > at .
Because is a deformation retract of , the inclusion map is a homotopy equivalence. Consequently, they possess isomorphic homotopy groups: This equivalence allows us to reduce the topology of infinite-dimensional path spaces directly to finite-dimensional Morse theory on .
With all the machinery of Morse theory that we require finally in place, we will now give a proof of the fundamental stability property for the homotopy groups of spheres. Specifically, this states that the homotopy groups are constant for all , a classic result that follows directly from the Freudenthal Suspension Theorem which we will now prove.
Let be a smooth real-valued function with minimum , such that each sublevel set is compact.
Lemma : If the set of minimal points is a manifold, and if every critical point >in has index , then for
First, observe that since is a smooth submanifold of , we can invoke the Tubular Neighborhood Theorem. This guarantees the existence of a tubular neighborhood of in and a smooth retraction map such that for all , making a retract of .
Furthermore, since is a Riemannian manifold, we can shrink appropriately so that for each point , its image and itself are joined by a unique minimal geodesic (a standard property in Riemannian geometry). By sliding uniformly along these unique geodesics from to , we obtain a smooth deformation retraction of onto . As the following picture below shows :
Now, let denote the unit -dimensional cube with boundary , where , and let be any relative map. We wish to show that is homotopic rel boundary to a map whose image lies entirely within .
Let be the maximum value of on . Let be chosen such that it is smaller than the minimum of on the closed complement (which exists because each sublevel set minus is compact).
Next, we choose a smooth function that approximates extremely closely, satisfying:The function is smooth on the compact region , and all its critical points are non-degenerate with index . Consequently, the manifold has the homotopy type of with attached cells of dimension from the fundamental theorem of finite dimensional Morse Theory.
Now consider our map: Since the dimension of the domain is strictly less than (), and the manifold is constructed by attaching cells of dimension , it follows from the that is homotopic within relative to to a map:
Finally, this image pair is contained entirely within . Since deformation retracts onto via our Riemannian geodesic flow, is homotopic within to a final map , proving that . The following diagram sums up what we have done in this proof.
Building on our foundation with the previous Lemma, we can now state the following theorem
Theorem : If the space of minimal geodesics from to is a topological manifold, and if every non-minimal geodesic from to has index , then the relative homotopy group is zero for .
The proof proceeds by approximating the path space using finite-dimensional smooth manifolds. Clearly, it is sufficient to prove that for arbitrarily large values of ~. As established in earlier chapters, the space contains a smooth manifold as a deformation retract, and the space of minimal geodesics is fully contained within this smooth manifold.
The energy function , when restricted to , almost satisfies the hypotheses of Lemma. The only technical difficulty is that ranges over the finite interval instead of the required interval ~. To fix this, we can let be any smooth diffeomorphism.
The composition now fully satisfies the hypotheses of the previous Lemma. Hence, the relative homotopy groups satisfy: which is zero for . This completes the proof of Theorem.
We can derive a powerful structural consequence of this theorem by examining the long exact sequence of relative homotopy groups for the pair :
From Theorem, we know that for ~. Inspecting the exact sequence around these vanishing relative groups leads directly to our main takeaway:
Corollary :
When is the sphere , the space of minimal geodesics can be naturally identified with the equator of and hence with itself. The index of all geodesics from the south pole to the north pole is given by , with being the index of the smallest non-constant geodesic. Thus, our index bound parameter is . Applying our corollary to the case of the sphere yields the celebrated :
Freudenthal Suspension Theorem : The suspension homomorphism is an isomorphism for .
Now, the following Stable property of the Homotopy groups is clear.
Stable Homotopy Property of Spheres : for , establishing the stability of the homotopy groups of spheres in the stable range.
Having established the foundational tools of algebraic topology and infinite-dimensional path spaces, we now calculate specific higher homotopy groups of spheres. A fundamental vanishing result governs the dimensions where the sphere dimension exceeds the homotopy dimension:
Theorem : For any integers and such that , the -th homotopy group of the -dimensional sphere vanishes:
This result follows directly from the stable range of homotopy groups of spheres (via the Freudenthal suspension theorem) combined with connectivity properties. Specifically, for , the difference , which implies that . Consequently, the sphere is simply connected, meaning its first homotopy group vanishes:
Through the structural stability and suspension isomorphisms, this vanishing propagates downward, yielding the general vanishing property for all dimensions below the sphere dimension.
We can also verify this vanishing property using tools from differential topology and Sard’s theorem, providing a geometric perspective on why higher-dimensional spheres cannot wrap around lower-dimensional ones non-trivially.
Therefore, any map with is homotopic to a constant map, confirming the vanishing theorem.
To study the homotopy groups of spheres further, it is essential to introduce a fundamental fiber bundle known as the .
We can understand the origin of this bundle geometrically: can be thought of as attaching a cell to . Now, can be identified with (or viewed as a quotient space of whose orbits are circles), and the boundary of the attaching cell is , which maps to this to . By viewing as , we obtain a natural map from to whose fibers are circles . A little bit of work shows that this map is locally trivial, yielding the fiber bundle:
From the machinery of algebraic topology established earlier, since the base space is paracompact and the projection is a locally trivial fiber bundle, we obtain a long exact sequence of homotopy groups:
Let us extract the relevant chunk of this exact sequence for lower dimensions:
We know from our earlier results that the homotopy groups of the 3-sphere in these dimensions vanish, specifically:
Substituting these vanishing terms into our exact sequence yields:
By exactness, the homomorphism between and is an isomorphism. Since the fundamental group of the circle is known to be the integers, , we immediately conclude that:
From stable homotopy theory, we know that the homotopy groups stabilize for large (specifically in the stable range where ). In particular, the homotopy groups stabilize starting from . Since we have just calculated that , this stabilization property yields:
This gives us the .
While exact sequences and path space approximations provide algebraic power, topology is fundamentally geometric. The allows us to view the homotopy groups of spheres through a completely geometric lens—translating abstract homotopy classes of maps into tangible submanifolds equipped with normal bundle trivializations.
Before establishing the main correspondence, we formalize the notions of framed submanifolds and framed cobordisms.
Let be a smooth, connected -dimensional manifold, and let be a closed, smooth -dimensional submanifold. The normal bundle of in is an -dimensional vector bundle over .
A of in is a choice of linearly independent sections of the normal bundle that span the normal space at every point . A pair consisting of a submanifold together with a choice of normal framing is called a of .
Let and be two -dimensional framed submanifolds of . We say and are if there exists a smooth, compact -dimensional submanifold with boundary such that:
Framed cobordism forms an equivalence relation. The set of equivalence classes of -dimensional framed submanifolds in is denoted by .
The fundamental insight of Lev Pontryagin and René Thom is that homotopy classes of maps into a sphere correspond bijectively to framed cobordism classes of submanifolds.
Theorem (Pontryagin-Thom Theorem) : For all integers and , there is a natural bijection between the homotopy classes of maps from to and the framed cobordism classes of , -dimensional submanifolds of
Equipped with the Pontryagin-Thom construction, we can now compute a non-trivial stable homotopy group of spheres completely geometrically. Specifically, we wish to calculate , which corresponds via stabilization to the 1st stable homotopy group .
By the Pontryagin-Thom theorem, this is isomorphic to the group of 1-dimensional framed cobordism classes in :
By the classification of compact, connected 1-dimensional manifolds without boundary, any such submanifold is diffeomorphic to the circle . Thus, we only need to study framed circles embedded in . To classify them up to framed cobordism, we divide the analysis into two natural cases: different framings on the same submanifold, and different submanifolds altogether.
Suppose we fix a single embedded circle .
Thus, up to framed cobordism, there are exactly two distinct framings that can be placed on a fixed 1-dimensional submanifold .
Suppose we have two different embedded circles, and , in .
Combining both cases, the classification of 1-dimensional framed submanifolds in up to cobordism reduces entirely to the homotopy classes of loops in . Since there are exactly two such classes, we arrive at the celebrated calculation:
It should be noted that this is not strictly a proof but his geometric approach visually explains why the 1st stable homotopy group of spheres collapses to .
Morse theory allows us to obtain cell decompositions of a manifold, while path spaces help us get a foothold on higher homotopy groups. Combined with foundational homotopy theory results, the Morse theory of and its path spaces provides a powerful machinery for establishing the stability of higher homotopy groups.
Toward the end, we explored the Pontryagin-Thom construction—a deeply geometric lens through which to view homotopy groups. In principle, one can through the Pontryagin-Thom theorem prove the Stable Homotopy of Spheres, making the underlying intuition far more transparent. However, doing so requires careful general position arguments from differential topology, such as the Thom-Jet transversality theorems, which we bypassed here.
Nevertheless, this perspective lets us catch a glimpse of the first stable homotopy group, a structure that is far from obvious using only homotopy exact sequences unless you bring out heavy artillery like the Serre or Adams spectral sequences.
Ultimately, the grand takeaway is this: to prove a general topological result, it is often vastly more fruitful to smooth it out and tackle it in the smooth category, where a much richer arsenal of geometric and analytic techniques is at your disposal.