Homotopy, Degree, Winding, and Covering Spaces

Two maps—or two field configurations described by maps—belong to the same homotopy sector when one can be continuously deformed into the other while remaining inside the declared class of admissible configurations. Different values of a homotopy invariant obstruct such a deformation. For maps between connected, closed, oriented manifolds of the same dimension, one such invariant is the integer degree; for a map from a circle to a circle, it is the winding number. A covering space makes the circle integer concrete: after lifting the phase to the real line, the lift returns shifted by an integer multiple of $2\pi$.

The qualification about admissibility is essential in QFT. Boundary values, regularity, finite-action conditions, excluded zeros or singularities, and the gauge transformations being quotiented all help define the configuration space. An integer is constant only along deformations that preserve those choices. This page develops the reusable mathematics and applies it to one $U(1)$ transition function on a two-patch sphere. It does not infer a stable solution, a superselection rule, or a physical charge from topology alone. Weinberg 1996, Volume II, § 23.1, pp. 422–423 illustrates how finite energy or Euclidean-action conditions turn field configurations into based maps and homotopy classes.

Homotopy setting and admissible configurations

The definitions require only continuous maps and assume no previous algebraic topology. The smooth degree formula later also uses elementary manifolds, orientation, and integration; Smooth Manifolds, Tangent and Cotangent Spaces, Tensors, and Lie Derivatives and Differential Forms, Integration, Orientation, and Stokes Theorem provide optional review for that one calculation rather than prerequisites for homotopy or winding.

Throughout,

• $I=[0,1]$ is the unit interval;
• $S^n$ has a chosen base point $s_0$ when based maps are discussed;
• $S^1$ and $U(1)=\{z\in\mathbb C:|z|=1\}$ carry the counterclockwise orientation;
• maps and homotopies are continuous unless the word smooth is present; and
• the degree of a map uses the displayed orientations of both source and target.

For a field theory, the symbol $\mathcal C_{\mathrm{adm}}$ will mean the configuration space after its boundary conditions, regularity, target, and gauge quotient have been specified. Different choices can produce different components.

Admissible deformations define homotopy sectors

Let $f_0,f_1:X\to Y$ be continuous maps. A homotopy from $f_0$ to $f_1$ is a continuous map

$$H:X\times I\longrightarrow Y, \qquad H(x,0)=f_0(x), \qquad H(x,1)=f_1(x).$$

If a subspace $A\subseteq X$ must remain fixed, a homotopy relative to $A$ also obeys

$$H(a,t)=f_0(a)=f_1(a) \qquad (a\in A,\ t\in I).$$

Taking $A=\{x_0\}$ gives a based homotopy. Leaving $A$ empty gives a free homotopy. The distinction records real information: fixed spatial boundary data lead naturally to relative homotopy, while a loop with a marked point leads to based homotopy.

Homotopy is an equivalence relation. A constant-in-$t$ homotopy gives reflexivity, $H(x,1-t)$ gives symmetry, and rescaling the first and second halves of $I$ concatenates two homotopies to give transitivity. The set of classes is written $[X,Y]$ in the free case; it is generally only a set, not a group.

This is different from saying that the spaces $X$ and $Y$ are homotopy equivalent. That stronger statement asks for maps $f:X\to Y$ and $g:Y\to X$ with

$$g\circ f\simeq\operatorname{id}_X, \qquad f\circ g\simeq\operatorname{id}_Y.$$

Two configurations are in the same homotopy sector precisely when an admissible one-parameter deformation joins them. If $\mathcal C_{\mathrm{adm}}$ is equipped with a topology whose continuous paths are exactly these deformations, the set of sectors is

$$\pi_0(\mathcal C_{\mathrm{adm}}).$$

Here $\pi_0$ denotes a set of path components. It has no automatic group law. When configurations are maps $X\to Y$, the allowed paths are the admissible homotopies specified above. Hatcher 2002, Chapter 0, pp. 1–3, PDF develops homotopy and homotopy equivalence from these definitions.

Based spheres produce homotopy groups

For $n\geq1$, the $n$th homotopy group of a based space $(Y,y_0)$ is

$$\pi_n(Y,y_0) = [(S^n,s_0),(Y,y_0)]_*.$$

The star means that maps and homotopies preserve the base point. Equivalently, one may use maps $(I^n,\partial I^n)\to(Y,y_0)$. For $n=1$, multiplication is concatenation of based loops and the group need not be Abelian. For $n\geq2$, the group is Abelian. A path between two base points in a path-connected space induces an isomorphism between the corresponding groups, but the isomorphism need not be canonical. For $\pi_1$, changing the path changes the result by conjugation; for higher groups, the dependence is described by the natural $\pi_1$ action on $\pi_n$. Accordingly, free homotopy classes of loops correspond to conjugacy classes in $\pi_1$, not generally to individual elements.

A based map $f:(X,x_0)\to(Y,y_0)$ induces

$$f_*:\pi_n(X,x_0)\longrightarrow\pi_n(Y,y_0), \qquad f_*[\alpha]=[f\circ\alpha].$$

Based-homotopic maps induce the same homomorphism, and a homotopy equivalence induces isomorphisms on all homotopy groups. For a free homotopy, moving the base point along its traced path supplies the corresponding change-of-base- point isomorphism; on $\pi_1$, this is the conjugation just described.

Some familiar integers are special cases:

$$\pi_1(S^1)\cong\mathbb Z, \qquad \pi_n(S^n)\cong\mathbb Z \quad(n\geq1).$$

The first integer counts winding; the second is degree. These examples must not be promoted to a claim that every homotopy sector has an integer label. Homotopy groups can be finite, non-Abelian in degree one, or much more complicated. Hatcher 2002, § 1.1, pp. 25–31, and § 4.1, pp. 340–343, PDF constructs the fundamental and higher groups and records the base-point action.

Covering spaces turn loops into endpoint data

A surjective map $p:\widetilde X\to X$ is a covering map if every $x\in X$ has an open neighborhood $U$ whose inverse image is a disjoint union

$$p^{-1}(U)=\coprod_{\alpha}U_\alpha,$$

with each restriction $p:U_\alpha\to U$ a homeomorphism. Such a $U$ is evenly covered. The local copies make two lifting facts possible:

1. a path in $X$ has a unique lift after its starting point in the fiber is chosen;
2. a homotopy has a unique lift after a lift of its entire initial map is chosen.

Thus a based loop in $X$ permutes the fiber $p^{-1}(x_0)$ by lifting and recording its endpoint. Homotopic loops have the same endpoint action. This is monodromy.

There is also an exact criterion for lifting a whole map. Let $p:(\widetilde X,\widetilde x_0)\to(X,x_0)$ be a covering and let $f:(Y,y_0)\to(X,x_0)$, where $Y$ is path-connected and locally path-connected. A based lift $\widetilde f:Y\to\widetilde X$ exists exactly when

$$f_*\pi_1(Y,y_0) \subseteq p_*\pi_1(\widetilde X,\widetilde x_0).$$

Here $f_*$ and $p_*$ are the induced homomorphisms on fundamental groups. Once $\widetilde f(y_0)=\widetilde x_0$ is fixed, the lift is unique. This criterion concerns a single-valued lift on all of $Y$; every path $I\to X$ still lifts after its starting point is chosen.

If $X$ is path-connected, locally path-connected, and semilocally simply connected, it has a universal cover. The last condition means that every point has a neighborhood whose loops become null-homotopic when regarded as loops in $X$. Under the same standard hypotheses, connected based coverings correspond to subgroups of $\pi_1(X,x_0)$; forgetting the chosen point over $x_0$ identifies conjugate subgroups. A covering projection induces isomorphisms on $\pi_n$ for $n\geq2$, while its effect on $\pi_1$ is the special injective map seen in the lifting criterion. Universal covers need not exist after these local hypotheses are dropped. Hatcher 2002, § 1.3, especially pp. 60–70, PDF proves the lifting statements and develops monodromy and the covering classification; Hatcher 2002, § 4.1, Proposition 4.1, p. 342, PDF gives the effect of a covering on higher homotopy groups.

The universal cover of the circle measures winding

Consider

$$p:\mathbb R\longrightarrow U(1), \qquad p(\alpha)=e^{i\alpha}.$$

Let $u:S^1\to U(1)$ and parametrize the domain by $e^{i\theta}$ with $0\leq\theta\leq2\pi$. After choosing the initial value, path lifting gives a continuous real function $\widetilde\phi$ such that

$$u(e^{i\theta})=e^{i\widetilde\phi(\theta)}.$$

Because the endpoints $\theta=0$ and $2\pi$ represent the same point of the domain, their lifts differ by a deck transformation. A deck transformation is a self-homeomorphism $d$ of the covering space satisfying $p\circ d=p$; for this cover they are precisely the translations $\alpha\mapsto\alpha+2\pi k$, with $k\in\mathbb Z$. Therefore

$$\widetilde\phi(2\pi)-\widetilde\phi(0)=2\pi\nu(u), \qquad \nu(u)\in\mathbb Z.$$

Changing the initial lift adds the same multiple of $2\pi$ everywhere, so $\nu(u)$ is unchanged. Homotopy lifting makes the endpoint difference constant during a homotopy. Conversely, if $u_0$ and $u_1$ have the same integer $\nu$, their lifts have the same quasiperiodicity, and

$$\widetilde H(\theta,t) =(1-t)\widetilde\phi_0(\theta) +t\widetilde\phi_1(\theta)$$

descends after exponentiation to a homotopy $H:S^1\times I\to U(1)$. Hence

$$[S^1,U(1)]\cong\mathbb Z:$$

for circle-valued maps, equal winding is both necessary and sufficient for homotopy.

If $u$ is smooth, the same integer has the global integral form

$$\nu(u) = \frac{1}{2\pi i} \int_{S^1}u^{-1}\mathrm du = \frac{\widetilde\phi(2\pi)-\widetilde\phi(0)}{2\pi}.$$

For $u_k(e^{i\theta})=e^{ik\theta}$, the lift is $k\theta$ and both formulas give $\nu(u_k)=k$. For circle self-maps, pointwise multiplication and composition obey different laws:

$$\nu(uv)=\nu(u)+\nu(v), \qquad \deg(u\circ v)=\deg(u)\deg(v).$$

Reversing the orientation of either circle, or replacing $u$ by $u^{-1}$, flips the sign. A nonzero-winding map has no single-valued global real phase on $S^1$: $\widetilde\phi$ is a lift on the parameter interval or on $\mathbb R$, not a function on the circle. Nakahara 2003, §§ 4.1–4.6, especially pp. 121–132 and 147–151 gives this lift construction, the circle degree, and the universal cover.

Degree counts oriented preimages

The circle result extends to equal-dimensional manifolds. Let $f:M^n\to N^n$ be a smooth map between connected, closed—compact and without boundary—orientable $n$-manifolds with chosen orientations. If $y\in N$ is a regular value, then $\mathrm d f_x:T_xM\to T_yN$ is an isomorphism at every $x\in f^{-1}(y)$. The inverse image is discrete and, by compactness, finite. Define

$$\varepsilon_x(f) = \begin{cases} +1,&\mathrm d f_x\text{ preserves orientation},\\[2pt] -1,&\mathrm d f_x\text{ reverses orientation}. \end{cases}$$

The degree theorem states that

$$\deg(f) = \sum_{x\in f^{-1}(y)}\varepsilon_x(f) \in\mathbb Z$$

is independent of the chosen regular value. The empty sum is zero. If $\Omega$ is an oriented top form on $N$ normalized by $\int_N\Omega=1$, the equivalent integral formula is

$$\deg(f)=\int_M f^*\Omega.$$

Without normalization, the right-hand side is divided by $\int_N\Omega$. In the fundamental-class language developed on Chains, Homology, Cohomology, and Exact Sequences, the same integer is defined by

$$f_*[M]=\deg(f)[N].$$

The integral gives a short smooth check of homotopy invariance. For a smooth $F:I\times M\to N$ with $F(0,x)=f_0(x)$ and $F(1,x)=f_1(x)$, orient $I\times M$ by $\mathrm dt$ followed by the orientation of $M$. Then $\partial(I\times M)=(\{1\}\times M)\sqcup-(\{0\}\times M)$. Since every $n$-form on $N^n$ is closed, Stokes’ theorem gives

$$\begin{aligned} \deg(f_1)-\deg(f_0) &= \int_M f_1^*\Omega- \int_M f_0^*\Omega\\ &= \int_{\partial(I\times M)}F^*\Omega\\ &= \int_{I\times M}\mathrm d(F^*\Omega) =0. \end{aligned}$$

The topological degree theorem extends the invariance statement to continuous maps. If $f:M\to N$ and $g:N\to P$ are maps between connected, closed, oriented $n$-manifolds, other immediate checks are

$$\deg(\operatorname{id}_M)=1, \qquad \deg(g\circ f)=\deg(g)\deg(f).$$

A map of nonzero degree must be onto, and reversing either chosen orientation changes the sign. Frankel 2012, § 8.3a, pp. 210–213 gives the normalized-form and regular-value formulations; Hatcher 2002, § 2.2, pp. 134–136, PDF proves homotopy invariance and the basic composition checks.

For $n\geq1$, degree is complete for maps $S^n\to S^n$: equal degree implies homotopy Hatcher 2002, § 4.2, Corollary 4.25, p. 361, PDF. It is not complete for arbitrary source and target. On the torus $T^2=S^1\times S^1$, compare

$$c(z,w)=(1,1), \qquad r(z,w)=(z,1).$$

Both maps have degree zero because neither is onto. But $c_*$ is trivial, whereas $r_*(a,b)=(a,0)$ on $\pi_1(T^2)\cong\mathbb Z^2$. A free homotopy could change an induced fundamental-group map only by the basepoint conjugation described above; no conjugate of a trivial homomorphism is nontrivial. Thus $c$ and $r$ cannot be freely homotopic. Equal integers are therefore sufficient only when a classification theorem says so.

The closed, oriented, equal-dimensional hypotheses are not decoration. Manifolds with boundary require relative data, noncompact manifolds require properness or conditions at infinity, and unoriented manifolds require a different invariant such as mod-two degree. A regular-value sign sum cannot be taken at a critical value.

Monopole transition data carry a winding obstruction

The controlled QFT-facing example uses only gluing data. Cover a sphere $S^2$ by contractible northern and southern patches $U_N$ and $U_S$. Their overlap is an equatorial band that deformation retracts to $S^1$. Fix increasing azimuthal angle $\phi$ as the equator orientation, take the global form of the structure group to be $U(1)$, and define the north-from-south transition function by

$$g_{NS}(e^{i\phi})=e^{in\phi}, \qquad n\in\mathbb Z.$$

Its lift is $n\phi$, so

$$\nu(g_{NS}) = \frac{1}{2\pi i} \int_{S^1}g_{NS}^{-1}\mathrm d g_{NS} =n.$$

This integer survives changes of local trivialization. With the convention that local fiber coordinates obey $v_N=g_{NS}v_S$, let $v'_N=h_Nv_N$ and $v'_S=h_Sv_S$, where $h_N:U_N\to U(1)$ and $h_S:U_S\to U(1)$. Then

$$g'_{NS} = h_Ng_{NS}h_S^{-1}$$

on the overlap, and therefore

$$\nu(g'_{NS}) = \nu(h_N|_{S^1}) +\nu(g_{NS}) -\nu(h_S|_{S^1}) =n.$$

Each restricted $h_N|_{S^1}$ or $h_S|_{S^1}$ extends over a disk, so it is null-homotopic and has winding zero. Consequently the integer classifies this two-disk clutching data: $n=0$ can be deformed to trivial gluing, while $n\neq0$ cannot. Reversing the equator orientation or using $g_{SN}=g_{NS}^{-1}$ flips the displayed sign.

This is the topological input to a $U(1)$ monopole bundle, but no magnetic-flux normalization or charge quantization has been assumed. Nakahara 2003, Example 9.7, p. 365, and §10.5.2, pp. 400–401 works out the transition map and its monopole interpretation. Whether a large transformation is quotiented as a redundancy, acts as a boundary charge, or changes an action phase belongs to the Symmetry and Gauge Theory volume, specifically Large Gauge Transformations and Topological Sectors.

What a topological label does not prove

Topological conclusion	Additional physical question
Different invariant values forbid an admissible homotopy.	Did the theory choose the same boundary conditions, regularity, and gauge quotient used to define that homotopy?
A nonzero class has a continuous representative.	Does a smooth finite-action solution of the field equations exist in that class?
A class cannot change along a path that stays admissible.	Can time evolution, tunneling, a singular event, or a boundary process leave that admissible space?
The configuration space has disconnected components.	Are they energetically stable, quantum mechanically superselected, or physically identified by gauge transformations?

Topology answers the left column. The dynamics, state space, boundary theory, and observable algebra answer the right. In particular, a field taking values in a vacuum manifold is a map, but a gauge field is generally a connection on a bundle; it is not automatically classified by a single map into the gauge group.

Common pitfalls

Calling every class an element of a group. The set $[X,Y]$ and the component set $\pi_0(Y)$ need not carry a natural group law. The group structure of $\pi_n(Y,y_0)$ uses based spheres and $n\geq1$.

Interpolating the winding exponent. The expression $e^{i(k+t)\theta}$ does not define a map on $S^1$ for a generic $0<t<1$, because its values at $\theta=0$ and $2\pi$ disagree. It is not a homotopy from winding $k$ to winding $k+1$.

Using a global argument for a nonzero-winding map. A smooth local phase exists, but a single-valued real function $\arg u$ on the whole circle would be a global lift and would force winding zero. Use $u^{-1}\mathrm du$ or the real-line lift instead.

Applying degree without its hypotheses. The integer formula above needs equal dimension, orientations, compactness, and no boundary. Changing those conditions changes the correct invariant and theorem.

Treating equal degree as a universal classification. Degree classifies sphere self-maps, not arbitrary maps. The torus example has equal degree but different induced fundamental-group maps.

Reading stability from topology. A homotopy obstruction forbids one kind of continuous deformation. It does not supply an energy functional, solve an equation of motion, or prohibit all quantum transitions.

Exercises

Each optional check is followed by a solution.

1. Homotopy of maps versus homotopy equivalence

State the difference between $f_0\simeq f_1$ for two maps $X\to Y$ and $X\simeq Y$ for two spaces.

Solution

The first statement requires one continuous map $H:X\times I\to Y$ joining $f_0$ to $f_1$. The second requires maps in both directions, $f:X\to Y$ and $g:Y\to X$, whose two composites are homotopic to the respective identity maps. Homotopic maps have the same source and target; homotopy-equivalent spaces need not be homeomorphic.

2. Why the obvious winding interpolation fails

Let $u_2(e^{i\theta})=e^{2i\theta}$ and $u_3(e^{i\theta})=e^{3i\theta}$. Why does $H(e^{i\theta},t)=e^{i(2+t)\theta}$ fail to define a homotopy? If the loops are instead regarded as maps into $\mathbb C$, what must happen in any deformation that changes their winding about the origin?

Solution

For $0<t<1$, the formulas at the two representatives of the same domain point differ:

$$H(1,t)\big|_{\theta=0}=1, \qquad H(1,t)\big|_{\theta=2\pi}=e^{2\pi it}\neq1.$$

So the formula does not descend from the interval to $S^1$. Winding is also homotopy invariant for loops in $\mathbb C^\times$. A deformation in $\mathbb C$ that changes winding must therefore pass through the origin, where the loop leaves $\mathbb C^\times$ and its winding is not defined.

3. Compute the winding of a power map

For $u_k:S^1\to U(1)$, $u_k(z)=z^k$ with $k\in\mathbb Z$, compute the winding from both the lifted endpoint and the integral formula.

Solution

With $z=e^{i\theta}$, choose the lift $\widetilde\phi_k(\theta)=k\theta$. Its endpoint shift is $2\pi k$, so the lift definition gives $\nu(u_k)=k$. Also,

$$u_k^{-1}\mathrm d u_k =ik\,\mathrm d\theta,$$

and hence

$$\frac{1}{2\pi i} \int_0^{2\pi}ik\,\mathrm d\theta =k.$$

Negative $k$ reverses orientation; $k=0$ is the constant map.

4. Check the transition-function invariant

With $g'_{NS}=h_Ng_{NS}h_S^{-1}$, prove that patch redefinitions extending over both hemispheres cannot change $\nu(g_{NS})$.

Solution

Winding is additive under pointwise multiplication and changes sign under inversion, so

$$\nu(g'_{NS}) = \nu(h_N|_{S^1}) +\nu(g_{NS}) -\nu(h_S|_{S^1}).$$

Each boundary loop $h_N|_{S^1}$ and $h_S|_{S^1}$ extends over its disk. A loop that extends over a disk is null-homotopic: contract the disk radially and compose the extension with that contraction. Both added windings are therefore zero, leaving $\nu(g'_{NS})=\nu(g_{NS})$.

Synthesis and continuations

Maps are deformable precisely when an admissible homotopy joins them. Degree and winding are integer-valued homotopy invariants under their stated hypotheses, so unequal values are obstructions. The universal cover $\mathbb R\to U(1)$ identifies winding with a lifted endpoint shift and proves that the integer completely classifies circle maps. General degree is not a complete classifier, and no topological label alone proves physical existence, stability, or superselection.

For the algebraic machinery behind fundamental classes and induced maps, continue to Chains, Homology, Cohomology, and Exact Sequences. For the curvature representative of the monopole integer, continue to Characteristic Classes and Chern–Weil Theory. For the developed physics of winding sectors, boundary conditions, and the status of large gauge transformations, continue to Large Gauge Transformations and Topological Sectors.

References

• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, § 8.3a, pp. 210–213. This gives an independent differential-geometric treatment of degree through normalized top forms and oriented regular preimages.
• Allen Hatcher, Algebraic Topology — Open PDF, Cambridge University Press, 2002, Chapter 0; §§ 1.1, 1.3, 2.2, and 4.1; and Corollary 4.25. Hatcher supplies proofs of the homotopy, covering-space, and degree results used here.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Institute of Physics Publishing, 2003, Chapter 4, pp. 121–151, Example 9.7, p. 365, and § 10.5.2, pp. 400–401. Nakahara gives an introductory account of circle lifts and homotopy groups followed by the monopole application.
• Steven Weinberg, The Quantum Theory of Fields, Volume II: Modern Applications, Cambridge University Press, 1996, §§ 23.1–23.2, pp. 422–436. These sections connect finite-energy or finite-action boundary conditions with based maps, winding sectors, vortices, and monopoles.