Spaces of knotted circles and exotic smooth structures

Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension $n$ that are homeomorphic. We prove that the spaces of smooth knots $Emb(S^1, N_1)$ and $Emb(S^1, N_2)$ have the same homotopy $(2n-7)$-type. In the 4-dimensional case this means that the spaces of smooth knots in homeomorphic 4-manifolds have sets $\pi_0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi_1$. The result about $\pi_0$ is well-known and elementary, but the result about $\pi_1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie-Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie-Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in $N$ does not depend on the smooth structure on $N$. Our results also give a lower bound on $\pi_2 Emb(S^1, N)$. We use our model to show that for every choice of basepoint, each of the homotopy groups $\pi_1$ and $\pi_2$ of $Emb(S^1, S^1\times S^3)$ contains an infinitely generated free abelian group.

For embeddings of the circle M = S 1 , we can use Theorem B to show that the homotopy type of T 2 Emb(S 1 , N) is independent of the smooth structure on N. The following theorem is Theorem 4.1 in the text.
Theorem C. Let N 1 and N 2 be smooth n-dimensional manifolds that are homeomorphic. Then the quadratic approximations T 2 Emb(S 1 , N 1 ) and T 2 Emb(S 1 , N 2 ) are homotopy equivalent.
Theorem A is then a consequence of Theorem C and known estimates of the connectivity of the approximation map Emb(M, N) → T 2 Emb(M, N). In fact, Theorem C implies a slightly stronger conclusion than what we stated in Theorem A. In particular, if N is 4-dimensional then the approximation map Emb(S 1 , N) → T 2 Emb(S 1 , N) is not just an isomorphism on π 0 and π 1 but is also an epimorphism on π 2 . Therefore one can use Theorem C to derive a lower bound on π 2 (Emb(S 1 , N)). We calculate some examples in Section 5: we show that the space Emb(S 1 , Σ 4 ) of knots in any homotopy 4-sphere Σ 4 is simply-connected (see Proposition 5.2) and we give an example of a 4manifold N such that the map Emb(S 1 , N) → Imm(S 1 , N) has infinitely generated kernels on π 1 and π 2 (see Corollary 5.7).
The outline of this paper is as follows. In Section 1, we review some results on tangent bundles and blow-ups that we will use in the later sections. Section 2 contains a discussion of spaces of immersions; this both illustrates our general strategy and provides results that we use later on. Section 3 is the center of the text, where we give a new description of the quadratic approximation T 2 Emb(M, N) that is valid for all M and N of any dimension. We return in Section 4 to the case where the source M = S 1 is the circle, to deduce our main results for spaces of knotted circles in general targets. The final Section 5 specializes further to the case where the target N is a smooth 4-manifold, to give more specific examples in Viro's original context. In particular, we show that for every choice of basepoint, each of the homotopy groups π 1 and π 2 of Emb(S 1 , S 1 × S 3 ) contains an infinitely generated free abelian group.

Blow-ups
The results in this section are valid for manifolds of all dimensions. Only in the last subsection shall we work out the example of the circle in sufficient detail for later use.
Notation. Let A be a submanifold of a manifold X. We will denote the spherical normal bundle of A by S A (X), and the spherical blowup of X at A by X \ A.
Recall that X \ A is a manifold with boundary whose interior is the complement X \ A and whose boundary is S A (X). There is a commutative diagram The spherical blowup is locally modeled as follows: take an inclusion U ⊂ V of linear spaces. Then there is an inclusion j : V \U → V , and a projection q : V \U → S U (V ). The blowup V \U is the closure of the image of the natural map ( j, q) : V \U −→ V × S U (V ).
Proposition 1.1. If B is a submanifold of another manifold Y , and if f : X → Y is a smooth map with f −1 B = A that induces (via the derivative) a fiberwise monomorphism between normal bundles, then it induces a smooth map X \ A → Y \ B.
The induced map X \ A → Y \ B is defined by using the restriction of the map f between the interiors X \ A and Y \ B, and the map induced by the derivative f between the spherical normal bundles on the boundaries. More details can be found for example in [AK14].
We will mostly be interested in the case when X = N × N is the product of a manifold N with itself, and A is the diagonal.
Notation. We will denote the blow-up Note that if N is a closed manifold, then the boundary of N [2] is S N (N × N), the spherical normal bundle of the diagonal in N × N, which can be identified with the spherical tangent bundle of N.
Notation. We will denote the spherical tangent bundle of N by S(τN) or just S(N).
Thus there is a canonical homeomorphism S(N) ∼ = S N (N × N). We identify S(N) with the boundary of N [2] . Note that the pair (N [2] , S(N)) has a canonical action by the group Z 2 .
Locally, we have the following situation: Example 1.2. In the case when M = R m is the local model, a linear transformation gives The involution is free: it is the antipodal action on S m−1 and trivial on all other factors. In this model, we have is a Σ 2 -homotopy equivalence. In fact, both of these spaces are Σ 2 -homotopy equivalent to S m−1 with the antipodal action.
The following simple proposition is one of the main technical results of this paper. To start with, we choose any tubular Σ 2 -neighborhood A such that Here and elsewhere we identify M with the diagonal of M × M. Let f : M → N be a homeomorphism, and let We repeat this process twice more and find tubular Σ 2 -neighborhoods C and D of the respective diagonals and end up with a chain Once this is set up, we consider the three inclusions Next, observe that there is also a zig-zag of Σ 2 -equivariant equivalences of diagrams and similarly there is a zig-zag of equivalences connecting the diagrams Remark 1.4. It seems likely that the assumption that M and N are closed manifolds can be relaxed.
Example 1.5. We need to understand the pair (N [2] , S(N)) in the case N = S 1 . The complement of the diagonal in the torus S 1 × S 1 consists of the ordered pairs of distinct points on the circle, and this is homeomorphic to S 1 ×] 0, 2π [ under the map that sends a pair of distinct points to the pair consisting of the first point and the angle to the second point (counter-clock-wise, say). The involution that interchanges the two points is given, in this model, by (z,t) → (z exp(ti), 2π − t), and it obviously extends to the spherical blowup, which is the cylinder S 1 × [ 0, 2π ]. Note that the involution interchanges the two boundary components via (z,t) → (z, 2π − t) for t ∈ {0, 2π} and acts on the central circle as (z, π) → (−z, π). To summarize, there is a homeomorphism Notice also that the blow-up (S 1 ) [2] is Σ 2 -equivariantly homotopy equivalent to the circle S 1 with the antipodal involution.

Linear approximation: immersions
In this section we point out that while in general Imm(M, N) is sensitive to the smooth structure on N, the space Imm(S 1 , N) is not. This is true for target manifolds N of all dimensions. In the case M = S 1 we obtain that there are equivalences where Λ denotes the free loop space functor and S(N) is, as usual, the sphere tangent bundle of N.
The tangent bundle of a smooth manifold N is not a topological invariant: Milnor [Mil63, Cor. 1] showed that there are smooth manifolds that are homeomorphic, but where one of them is parallelizable, and the other one is not. In other words, there are smooth structures on some topological manifold that afford non-isomorphic tangent bundles.
On the other hand, the sphere bundle is to some extent a topological invariant. The following result is a corollary of theorems of Thom [Tho52, Cor. IV.2] and Nash [Nas55]. It also follows from our Proposition 1.3.
Proposition 2.1. If smooth manifolds M and N are homeomorphic, then the total spaces of the spherical tangent bundles S(M) and S(N) are homotopy equivalent.
Theorem 2.2. The homotopy type of the space Imm(S 1 , N) of immersion of the circle into a smooth manifold N does not depend on the smooth structure of N.
Proof. We saw that the space Imm(S 1 , N) is homotopy equivalent to ΛS(N). Now the result follows from Proposition 2.1.
Goodwillie and Klein [GK15] have shown that the connectivity of the map to the k-th layer in the Goodwillie-Weiss tower is at least k(n − m − 2) + 1 − m. Recall: a map is called cconnected if all of its homotopy fibers are (c − 1)-connected. In particular, the Goodwillie-Klein result implies the much more elementary fact that the inclusion Emb(M, N) → Imm(M, N) is (n − 2m − 1)-connected. It follows for M = S 1 that the inclusion Emb(S 1 , N) → Imm(S 1 , N) is (n − 3)-connected. Theorem 2.2 implies that the homotopy (n − 4)-type of Emb(S 1 , N) does not depend on the smooth structure of N. In the following sections, we shall roughly double this range.
We end this section with a couple of elementary observations about the set π 0 of components and the fundamental groups π 1 of Imm(S 1 , N). Let n be the dimension of N. The bundle map S(N) → N is (n − 1)-connected. It follows that there is an (n − 2)-connected map Assuming n 4, we have isomorphisms π i Imm(S 1 , N) ∼ = π i Λ(N) for i = 0, 1. Using well-known facts about the homotopy groups of π i Λ(N), we obtain the following proposition.
Proposition 2.3. Let N be a connected smooth manifold of dimension n 4. Then the set of components of the space Imm(S 1 , N) is in natural bijection with the set of conjugacy classes of elements in the fundamental group of N. If N is simply-connected, then the fundamental group of the space of immersions is isomorphic to π 2 N ∼ = H 2 (N; Z).

Quadratic approximations
In this section, we will give a new description of the quadratic approximation T 2 Emb(M, N) that is valid for all M and N.
The first order (a.k.a. linear) approximation to the space Emb(M, N) of embeddings M → N is given by the space Mono(τM, τN) of monomorphisms of tangent bundles [Wei99], and the corresponding approximation map Emb(M, N) → Mono(τM, τN) is induced by differentiation. To understand the quadratic approximation we need to study the behavior of maps on pairs of points.
Recall that we have two canonical ) as the space of commutative diagrams where the vertical arrows are canonical. In other words, there is a pullback square Notice that the boundary inclusion S(M) → M [2] is a Σ 2 -cofibration: it is a cofibration and the group Σ 2 acts freely on the complement of the image. It follows that the right vertical map in (3.1) is a fibration, and that the square diagram (3.1) it is both a strict and a homotopy pullback.
Lemma 3.1. The homotopy type of for a fixed source M, only depends on the homeomorphism type of N.
is an equivalence, the space is homotopy equivalent to the homotopy pullback of the diagram Clearly this homotopy pullback only depends on the homeomorphism type of N.
There is an evident commutative diagram This diagram induces a natural map.
There also exists a natural map.
Indeed, for an embedding f : Therefore, we can use Proposition 1.1 to produce a map: the blow-up of f × f at the diagonal.
Next, we claim that there is a commutative diagram To define this diagram we have to specify the top horizontal and the left vertical maps. The top horizontal map is the composition of the following maps, each one of which is obvious The left vertical map is the following composition of obvious maps It is an easy exercise to check that with these definitions the diagram (3.4) commutes. This diagram gives rise to a natural map Lemma 3.2. The following diagram commutes Proof. Let f : M → N be an embedding. Our task boils down to the question whether the diagram is the blowup of f × f at the diagonal, and the unlabelled arrows are the obvious ones. The diagram is commutative by definition of the map f [2] .
The commutative square in Lemma 3.2 is not a (homotopy) pullback in general. But it is in some important cases: Lemma 3.3. The commutative square in Lemma 3.2 is a homotopy pullback if M is the disjoint union of at most two copies of R m .
Proof. Let M ∼ = k × R m , where k is a finite set with k elements, and analyze the commutative square in this case. Our goal is to show that when k 2 the square is a homotopy pullback.
In the case the set k = 0 is empty, all the spaces involved are contractible and there is nothing to prove.
Next, suppose that k = 1 is a singleton, so that we have a homeomorphism M ∼ = R m . In this case the map Emb(R m , N) → Mono(τR m , τN) is an equivalence, so we need to show that the map is a homotopy equivalence. This is equivalent to showing that the diagram is a homotopy pullback square. The boundary inclusion S(R m ) → (R m ) [2] is a Σ 2 -equivariant homotopy equivalence (see Example 1.2). Therefore, both horizontal arrows are homotopy equivalences, and then the square must be a homotopy pullback.
Lastly, let us suppose that the set k = 2 consists of two points. Let M denote k × R m . There is an embedding of k into M, sending each point of k to the origin of the corresponding copy of R m . This embedding gives rise to the following diagram, where we have have used the abbreviation We want to prove that the front face is a homotopy pullback. For this it is enough to prove that the left, back, and right faces are each a homotopy pullback.
It is a standard fact that the left face is a homotopy pullback for M = k × R m , for any k: the difference between an embedding of R m and an embedding of its center is given by a framing of the tangent space at the center, and it is the same for mere immersions.
Note that since k is a zero dimensional manifold, S ∆ (k) = Ø and k [2] = k × k \ k. In particular, for k = 2, we have k [2] ∼ = Σ 2 as Σ 2 -sets. Then, by inspection, the back face is homeomorphic to the square Clearly the horizontal arrows are equivalences, and so we have a homotopy pullback square.
As for the right face, the general formula gives that in the case M = 2 × R m there is a homeomorphism and then This maps to Also, we have S(M N) ∼ = S(M) S(N), so that and the boundary inclusion of S(2 × R m ) into (2 × R m ) [2] becomes the evident map into the summand on the right: 2×boundary inclusion of S(R m ) → (R m ) [2] .
Together, we see that the right face becomes We proved that the map is a homotopy equivalence when we considered the case k = 1 (see (3.6)). It follows that the last square is a homotopy pullback.
For general M, we have the following result: is, by definition (3.1), itself a homotopy pullback of functors each one of which is easily shown to be of degree at most 2.

Embeddings of the circle
For a general source manifold M, we see no reason why the homotopy type of the quadratic approximation T 2 Emb(M, N) should be independent of the smooth structure on N. In this section, we will specialize to the case M = S 1 , so that the embedding spaces are spaces of knots. The target manifold N can still be arbitrary of dimension at least 4.
We are ready to state and prove our main theorem.
Theorem 4.1. The homotopy type of the space T 2 Emb(S 1 , N) does not depend on the smooth structure of the manifold N.
Proof. By Theorem 3.5, and together with the equivalence Imm(S 1 , N) ΛS(N), we already know that the space T 2 Emb(S 1 , N) is equivalent to the homotopy pullback of the following diagram.
We claim that the homotopy type of this diagram (4.1) is determined by the homotopy type of the diagram S(N) → N [2] → N × N. By Proposition 1.3, the latter is determined by the homeomorphism type of N and is independent of the smooth structure.
Because the lower right corner of the diagram (4.1) is defined as a pullback, that entire diagram is determined by the following diagram.
Using the fact that S(S 1 ) ∼ = Σ 2 × S 1 , we may rewrite this diagram as follows where f is induced by the squaring map, g is induced by the inclusion S(N) → N [2] and all the other maps should be self-evident. It is clear that the homotopy type of this diagram is determined by the homotopy type of the diagram S(N) → N [2] → N × N, and therefore so is the homotopy limit of this diagram, which is T 2 Emb(S 1 , N).
Corollary 4.2. The homotopy (2n − 7)-type of the space Emb(S 1 , N) does not depend on the smooth structure on the manifold N.
Proof. The approximation map Emb(M, N) → T 2 Emb(M, N) to the second layer in the Goodwillie-Weiss tower is (2(n − m − 2) + 1 − m)-connected. For the embeddings of the circle M = S 1 , this shows that the approximation map Emb(S 1 , N) → T 2 Emb(S 1 , N) is (2n − 6)-connected, so that both spaces share the same homotopy (2n − 7)type. Therefore, the theorem implies the corollary.
To end this section, we will describe the homotopy fiber of the map  Proof. By Theorem 3.5 and/or diagram (4.2), the homotopy fiber that we are interested in is equivalent to the total homotopy fiber of the following diagram The calculation of the total fiber is pretty straightforward. If there is any subtlety, it has to do with basepoints and the (lack of) dependence on the choice of basepoint. We will calculate the total fiber by first taking fibers in the horizontal direction and then the vertical direction.
Let F be the homotopy fiber of the map N [2] → N × N over the basepoint (x, x). Because the inclusion N × N \ N → N [2] is a homotopy equivalence, the space F is equivalent to the homotopy fiber of the inclusion N × N \ N → N × N. There is a map of (horizontal) fibration sequences, and computing homotopy fibers vertically, we see that F is homotopy equivalent to the homotopy fiber of the inclusion N \ {x} → N. In particular, the space F is 2-connected.
The total fiber that we are interested in is equivalent to the fiber of the following map Equivalently, we can write this map as follows, using S(S 1 ) ∼ = Σ 2 × S 1 : Note that the basepoint of Map Σ 2 (S(S 1 ), F) is not a constant map. (There is no constant Σ 2 -equivariant constant map into F, since the action of Σ 2 on F is free.) Rather, the basepoint is a map that is constant on each connected component of S(S 1 ) ∼ = Σ 2 × S 1 . It sends one copy of S 1 to the point (x, u) and the other copy to (x, − u). Let us explain how we think of the two points (x, ± u) as points in F. Initially (x, u) was defined to be the chosen basepoint of S(N). Since we have an inclusion S(N) → N [2] , the points (x, ± u) can also be thought as points of N [2] . In fact, they are points in the fiber of the map N [2] → N × N over the point (x, x). Therefore they naturally define points in the homotopy fiber of same map, which is F. It follows that the homotopy fiber of the map (4.3) is the space of equivariant maps from (S 1 ) [2] to F that take one path component of the boundary of (S 1 ) [2] to (x, u) and the other path component to (x, − u). Notice that this map from the boundary of (S 1 ) [2] to F can be extended to a Σ 2 -equivariant map from all of (S 1 ) [2] to F, because (S 1 ) [2] is two-dimensional and F is two-connected. In other words the fiber of (4.3) is not empty.
Next, we would like to stabilize. Let ΣF be the unreduced suspension of F. We use the unreduced suspension because F does not have a Σ 2 -equivariant basepoint. Let Ω ΣF be the space of paths in ΣF from the "south pole" to the "north pole." (By our convention, the south pole is the basepoint of ΣF.) There is a natural map F → Ω ΣF that is 5-connected because the space F is 2-connected. (This is a version of the Freudenthal suspension map for unpointed spaces.) It follows that the fiber of the map (4.3) is connected to the fiber of the map Map Σ 2 ((S 1 ) [2] , Ω ΣF) → Map Σ 2 (S(S 1 ), Ω ΣF) (4.4) by a 3-connected map.
Next, we claim that one can replace Ω with the usual loop space Ω in this map. To see this, observe that the homotopy fiber above can be identified with the space of Σ 2 -equivariant maps from (S 1 ) [2] × I to ΣF that agree with a prescribed map on the subspace The prescribed map is defined as follows. On S(S 1 ) × I it is the composite S(S 1 ) × I → F × I → ΣF, where the first map is determined by the basepoint map S(S 1 ) → F and the second map is the canonical quotient map. On the components of (S 1 ) [2] × ∂ I the prescribed map is constant with image the south pole and north pole, respectively. We claim that the prescribed map is Σ 2 -equivariantly homotopic to the constant map into the south pole: this claim follows from the fact that the basepoint map S(S 1 ) → F can be extended to a Σ 2 -equivariant map from (S 1 ) [2] to F, because the space F is 2-connected. It follows in turn that the homotopy fiber of the map (4.4) is equivalent to the space of pointed Σ 2 -equivariant maps from (S 1 ) [2] × I to ΣF that agree with the trivial map on the indicated subspace. This mapping space is the homotopy fiber of the map which is obtained from (4.4) by replacing the functor Ω with the ordinary loop space functor Ω. Note that now the basepoint of Map Σ 2 (S(S 1 ), Ω ΣF) is the constant map into the trivial loop.
(4.5) by a 4-connected map. It follows that the homotopy fiber that we are interested in is connected to the last homotopy fiber by a 3-connected map. Now the homotopy fiber is taken over the usual basepoint, given by the constant map. The homotopy fiber is equivalent to the space Map * Σ 2 (C, ΩQ ΣF).
It remains to identify the space ΣF with the Thom space (ΩN) τ . Recall that F denotes the homotopy fiber of the canonical map N Passing to its (horizontal) homotopy cofibers, we get an equivalence (ΩN) τ − → ΣF .

Applications to the fourth dimension
Let N = N 4 be a smooth connected 4-manifold. The tangent bundle of an oriented 4-manifold is determined by its topology (see [HH58] and [DW59]): Any oriented 4-plane bundle over a 4-dimensional complex is determined by its second Stiefel-Whitney class w 2 , its first Pontryagin class p 1 , and its Euler class e. For the tangent bundle, these classes are all topological invariants. This also implies Proposition 2.1 in dimension 4.
In this section, we work out the implications of our general results for the spaces Emb(S 1 , N 4 ): Corollary 5.1. The homotopy 1-type of the space Emb(S 1 , N 4 ) does not depend on the smooth structure on the 4-manifold N 4 .
This allows us to compute the set of components (isotopy classes of embeddings), which was known before, and all the fundamental groups of the components, which is new, from the topology alone. We also get a lower bound on π 2 .
Let us start with π 0 . If M = S 1 and N 4 is any 4-manifold, then the map Emb(S 1 , N 4 ) → Imm(S 1 , N 4 ) is 1connected, so that it induces a bijection between the sets of (path) components and an epimorphism on fundamental groups. For the set of components, we have Proposition 2.3, and get π 0 Emb(S 1 , N 4 ) ∼ = π 0 Imm(S 1 , N 4 ) ∼ = π 0 ΛS(N) ∼ = π 0 ΛN 4 , and that set is in natural bijection with the set of conjugacy classes of elements in the fundamental group π 1 N 4 . Note that this set only depends on the homotopy type of the 4-manifold N 4 .
We can now turn to π 1 and π 2 . For the moment, let us assume, for simplicity, that the manifold N 4 is simplyconnected. Then the space Imm(S 1 , N 4 ) is connected and the fundamental group of that space is H 2 (N 4 ; Z) by Proposition 2.3. Consequently, it is known that the space Emb(S 1 , N 4 ) is path connected and that the fundamental group (and the first homology group) of that space surjects onto the abelian group H 2 (N 4 ; Z). Corollary 5.1 lets us substantially improve on this estimate.
Proof. If Σ 4 = S 4 happens to be the standard 4-sphere, then the statement is known to be true: the embedding space Emb(S 1 , S 4 ) is simply-connected (see [Bud08,Prop. 3.9], for instance): the embedding space Emb(S 1 , S 4 ) has the same homotopy 1-type as the space of linear embeddings, the Stiefel manifold SO(5)/ SO(2). In general, Freedman has shown that every homotopy 4-sphere Σ 4 is homeomorphic to the standard 4-sphere S 4 , and then our Corollary 5.1 implies the result.
As another application of our results, we will now show that there are 4-manifolds N for which the inclusion Emb(S 1 , N) → Imm(S 1 , N) has a (very) non-trivial kernel on π 1 and π 2 : Example 5.4. Let us consider the space Emb(S 1 , S 3 × S 1 ). Because the target is parallelizable, we have an equivalence Imm(S 1 , S 3 × S 1 ) Λ(S 3 × S 3 × S 1 ). (5.1) It follows that the space Imm(S 1 , S 3 × S 1 ), and therefore also the space Emb(S 1 , S 3 × S 1 ) has a countably infinite number of connected components, indexed by the map induced between the fundamental (or first homology) groups. Notice that the path components of Imm(S 1 , S 3 × S 1 ) are homotopy equivalent to each other. We also have an analogous statement for the homotopy fibers of the map from T 2 Emb(S 1 , S 3 × S 1 ) to T 1 Emb(S 1 , S 3 × S 1 ) Imm(S 1 , S 3 × S 1 ) over different path components: Proposition 5.5. The homotopy type of the homotopy fiber of the map is the same for every choice of basepoint. In all cases, the homotopy fiber is equivalent to the space of equivariant pointed maps, where F is the homotopy fiber of the inclusion S 3 × S 1 \ (1, 1) → S 3 × S 1 . The action of the group Σ 2 on F is defined via the action on S 3 and S 1 that sends an element of S 1 or S 3 to its inverse as a complex number or quaternion.
Proof. It is enough to verify the proposition for a choice of one basepoint in each path component. We know that the following maps all induce a bijection on π 0 .
We want to choose a convenient set of representative basepoints. For the purpose of this proof, let us identify S 1 with the circle of unit complex numbers and S 3 with the unit quaternions. The point (−1, −1) is our basepoint for S 3 × S 1 . Let i : C → H be the inclusion of the complex numbers into the quaternions. Let α n : S 1 → S 3 × S 1 be the map defined by α n (z) = (i(z), z n ). Then {α n | n ∈ Z} is a complete set of representatives of the path components of Emb(S 1 , S 3 × S 1 ) and therefore their images give a complete set of representatives of the path components in the other spaces, too. We will show that the homotopy fibers of the map (5.2) are pairwise homotopy equivalent for all these basepoints.
We have seen in Theorem 3.5 and/or diagram (4.2), that the homotopy fiber of (5.2) is equivalent to the total homotopy fiber of the following square diagram.
To remember that the circle S 1 with the antipodal action arose as the middle of (S 1 ) [2] , we continue denoting the elements of S 1 as pairs (z, −z), where z is a unit complex number. The vertical maps in the diagram are induced by the canonical Σ 2 -equivariant maps S(S 1 ) ∼ = Σ 2 × S 1 → S 1 . Our task now is to compare the total fibers of the diagram (5.4) with basepoints in Map Σ 2 ( S 1 , (S 3 × S 1 ) [2] ) given by maps where n ∈ Z. ( Step 2) Notice that the boundary of (S 3 × S 1 ) [2] is not in the image of any of the maps that serve as basepoints of the mapping space Map Σ 2 ( S 1 , (S 3 × S 1 ) [2] ). It follows that in diagram (5.4) we may replace the space (S 3 × S 1 ) [2] with the homotopy equivalent space (S 3 × S 1 ) 2 \ S 3 × S 1 . So we are now interested in the total homotopy fiber of the following diagram.
Step 3) Our last step is to simplify further the total homotopy fiber of (5.6), using the fact that S 3 × S 1 is a Lie group.
Suppose G is a Lie group. Let G denote the underlying space of G, equipped with the Σ 2 -action that sends an element to its inverse. There is a fibration sequence G g →(g,g) Proposition 5.6. Both π 1 and π 2 of each homotopy fiber of the map Emb(S 1 , S 3 × S 1 ) −→ Imm(S 1 , S 3 × S 1 ) are abelian and contain an infinitely generated free abelian group. In particular, they are not finitely generated.
As for the Σ 2 -homotopy type of the homotopy fiber F, the space S 3 × S 1 \ (1, 1) is homotopy equivalent to S 3 ∨ S 1 , where we can take the wedge point to be (−1, −1). Let us recall that the group Σ 2 is acting on S 1 and S 3 by taking each element to its inverse (as elements in C or H). The Whitehead product fibration Σ(ΩA∧ΩB) → A∨B → A×B shows that we have equivalences where the last equivalence comes from the James splitting: the space ΣΩS 3 is homotopy equivalent to the wedge sum S 3 ∨ S 5 ∨ S 7 . . .. The action of the group Σ 2 on the indexing set Z sends n to its inverse −n.
As for the space S 1 × S 1 /S 1 , it fits into a cofibration sequence of pointed Σ 2 -equivariant maps. By adjunction, any such map f is described uniquely by an element in π 1 (Σ 2+ ∧ S 1 ), which is the free group on two generators, say a and σ (a), where σ is the non-trivial element of the group Σ 2 . The map f in question is a · σ (a).
Taking the homotopy long exact sequence, and focusing on π 2 and π 1 , we obtain the following exact sequence.
Here Z(n) denotes a copy of the group Z corresponding to the index n. The homomorphism in the middle splits as a product of a homomorphism Z(0) → Z(0), which we do not need to determine, and, for each n > 0, the homomorphism Z(n) × Z(−n) → Z(n) × Z(−n) that sends a pair (i, j) to the pair (i + j, i + j).
The group π 2 Map * Σ 2 S 1 × S 1 /S 1 , F is abelian, because it is a π 2 , and by exactness of the sequence above, it surjects onto the kernel of the middle homomorphism, which obviously contains an infinitely generated free abelian group. Therefore, the group π 2 Map * Σ 2 S 1 × S 1 /S 1 , F itself also contains an infinitely generated free abelian group. As for the fundamental group π 1 Map * Σ 2 S 1 × S 1 /S 1 , F , we first note that the exact sequence implies that it is abelian as well, as the quotient of an abelian group. And the cokernel of the middle homomorphism, which is isomorphic to π 1 Map * Σ 2 S 1 × S 1 /S 1 , F , also contains an infinitely generated free abelian group.
Corollary 5.7. For j = 1 and j = 2, the kernel of the homomorphism π j Emb(S 1 , S 3 × S 1 ) −→ π j Imm(S 1 , S 3 × S 1 ) is abelian and contains an infinitely generated free abelian group. In particular, the kernels are not finitely generated.
We have also seen that the group π 1 Emb(S 1 , S 3 × S 1 ) contains an infinitely generated free abelian group. We refer to the Budney's and Gabai's more recent preprint [BG] for more information on these fundamental groups. Our methods allow us to obtain information on higher homotopy groups as well: Corollary 5.8. The group π 2 Emb(S 1 , S 3 × S 1 ) contains an infinitely generated free abelian group. In particular, it is not finitely generated.
It would be interesting to see a calculation showing an example of a simply-connected 4-manifold N for which the map Emb(S 1 , N) → Imm(S 1 , N) has a non-trivial kernel on π 1 . (Moriya's recent preprint [Mor] contains restrictions that apply.) It is easy to show that the homotopy fiber of the map T 2 Emb(S 1 , N) → Imm(S 1 , N), and therefore also of the inclusion Emb(S 1 , N) → Imm(S 1 , N), has non-trivial π 1 for many manifolds N, including simplyconnected ones. But we have not analyzed the long exact sequence in homotopy in enough detail to show that the map from the homotopy fiber to T 2 Emb(S 1 , N) is non-zero on π 1 for some simply-connected N.