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Algebraic topology translates questions about topological spaces and continuous maps into questions about groups and other structures from algebra. A topological invariant compresses the enormous amount of data in a space into something one can calculate. For example, the fundamental group records which based loops can be continuously deformed into one another. A filled disk has trivial fundamental group, whereas a circle has fundamental group $\mathbb{Z}$: an integer remembers how many times a loop goes around.

This compression also respects maps. By functoriality of the fundamental group, a continuous map preserving the chosen basepoints induces a group homomorphism; successive maps induce successive group homomorphisms. Thus geometry supplies arithmetic constraints. A proposed homotopy equivalence must induce an isomorphism, so incompatible fundamental groups rule it out before anyone attempts to construct the deformation.

The beauty is that a whole annulus can retain exactly the same homotopy type as its inner circle. Its thickness disappears under a deformation retraction, while the obstruction to shrinking a loop survives. Equal fundamental groups alone do not establish a homeomorphism or a homotopy equivalence between arbitrary topological spaces.

This turns a visual question into a reusable test. Suppose a continuous map sends the inner circle of an annulus once around another circle. Could it extend over a filled disk spanning that boundary? Such an extension would supply a homotopy shrinking the image loop to a point. Its winding number would have to change from one to zero, which homotopy invariance of winding number forbids. Algebra rules out the extension even when no particular formula for it has been proposed.

Normalize the section to the sphere bundle. Scaling it from zero to unit length gives a homotopy within the disk bundle from the zero section to this sphere-valued section. The image of the relative Thom class in absolute cohomology restricts to zero on the sphere bundle. Homotopy invariance therefore makes its zero-section pullback vanish.

// Target: geometry-and-topology.bigb

A finite list of function values does not determine the winding number of the intervening image contour. For example, sample $\gamma (t)=e^{it}$ only at $t_j=2\pi j/4$, $j=0,1,2,3$. The polynomials $f(z)=z$ and $g(z)=z^5$ agree at all four sampled points, because $\gamma (t_j{)}^4=1$. Joining those values by straight segments draws the same square for both. Yet their true winding numbers are one and five, and their enclosed zero multiplicities differ by four.

Refining samples can help resolve a given example, but an asserted root count needs a bound controlling what happens between them. One sufficient certificate is to enclose the image of each boundary subarc and its interpolating segment in a convex set that excludes zero. Straight interpolation then gives a homotopy avoiding zero, so the polygon and true image contour have the same winding number. A plot that merely looks closed and avoids zero at its sampled points is not that certificate.

A reduced contravariant homotopy functor on based CW complexes satisfying the wedge-to-product axiom and the Mayer-Vietoris weak-pullback gluing axiom is represented by $[X,Y{]}_{\ast }$ for a CW complex $Y$. Half-exactness alone is not the full hypothesis. A reduced generalized cohomology theory supplies these axioms; suspension compatibility assembles its representing spaces into an Omega-spectrum.

On a compact smooth manifold without boundary, every smooth self-map sufficiently $C^1$-close to a diffeomorphism is a diffeomorphism. Reducing to a map near the identity, invertibility of its differential follows uniformly by compactness. It is then a proper local diffeomorphism, hence a finite covering map. Sufficient $C^0$ closeness gives a homotopy to the identity on each component; its induced map on the fundamental group is surjective. The covering therefore has one sheet on each component, proving the assertion. This lets a nearby Lagrangian submanifold transverse to cotangent fibers be represented by a single global differential one-form graph.

A collar neighborhood identifies a neighborhood of the boundary with $\partial W\times [0,\epsilon )$, with the boundary at parameter zero. It allows a manifold's boundary to be pushed slightly inward by a homotopy.

A contraction is a homotopy from the identity map of $X$ to a constant map. Its existence says that $X$ is a contractible space. For a tree, choosing a root and moving each point along its unique finite path to that root gives a contraction, with continuity understood in the weak topology of a CW complex.

A deformation retraction of $X$ onto a subspace $A$ is a homotopy $H:X\times [0,1]\to X$ such that $H(x,0)=x$, $H(x,1)=r(x)\in A$, and $H(a,t)=a$ for every $a\in A$ and every $t$. Thus the retract is fixed pointwise throughout. With inclusion $i:A\hookrightarrow X$, the equations $r\circ i={\mathrm{id}}_A$ and $i\circ r\simeq {\mathrm{id}}_X$ exhibit a homotopy equivalence. See Hatcher, Algebraic Topology, Chapter 0, pp. 2–3.

Take the closed annulusAn exact deformation retraction onto its inner circle isThe denominator is nonzero everywhere on the annulus. If the starting radius is $\rho =\Vert x\Vert$, its radius at time $t$ is $(1-t)\rho +t$. This lies between $1$ and $\rho \le 2$, verifying that the entire homotopy stays in the original annulus. When $\rho =1$, the point is stationary, as required.

For $x=(0,2)$, times $0,1/2,1$ give $(0,2),(0,3/2),(0,1)$. A radius-$3/2$ loop similarly becomes a radius-$5/4$ loop halfway through, then the unit circle. Its angle and winding number are preserved. This is useful because computing the fundamental group of the annulus reduces to computing that of the circle; the explicit homotopy equivalence establishes why the answer transfers.

A nonzero topological degree obstructs smooth unwinding with fixed boundary data, but does not by itself give a stable finite size. In three dimensions a two-derivative energy scales as $R$ and can decrease as a configuration shrinks at fixed degree. The four-derivative Skyrme term scales as $R^{-1}$ and can balance this tendency. Derrick theorem addresses the energetic issue; a singular limiting configuration can evade smooth homotopy conservation.

Although topological degree classifies maps $S^n\to S^n$ up to homotopy, it is not a complete invariant for other manifolds. On the torus, the identity and the integer shear $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ both have degree one, but induce distinct maps on $H_1(T^2;\mathbb{Z})$ and cannot be homotopic.

For a map $f:S^m\to S^m$, the degree is the integer by which $f_{\ast }$ multiplies a chosen generator of $H_m(S^m;\mathbb{Z})$. It is invariant under homotopy.

A graded algebra equipped with a degree-one differential satisfying $d(ab)=d(a)b+(-1{)}^{|a|}ad(b)$ and $d^2=0$. Its cohomology inherits a graded algebra structure. In rational homotopy, a Sullivan minimal model is a free graded-commutative differential graded algebra with decomposable generator differentials.

Choose a complex structure $J$ on ${\mathbb{R}}^{2m}$, with $J^2=-I$. Its induced map on $\mathbb{R}{P}^{2m-1}$ has no fixed point: $Jx=\lambda x$ for a real nonzero vector would give ${\lambda }^2=-1$. The maps $\cos tI+\sin tJ$, for $0\le t\le \pi /2$, give a homotopy from the identity to $J$, because they are invertible throughout. Thus this fixed-point-free map is homotopic to the identity.

Double a compact orientable genus-zero topological surface with $g+1$ boundary circles. The resulting closed surface has genus $g$. The involution exchanging the halves fixes the seam circles and has local collar form $(s,t)\mapsto (s,-t)$. Compose it with a flow rotating each seam coordinate by a small nonzero amount, with speed supported in its collar. Equality with the initial point would force $t=0$, where the rotation is nonzero; off the collars the original reflection already has no fixed points. The new map is fixed-point-free and homotopic to the reflection through the flow. This gives a surface example in every genus where having fixed points is not a homotopy invariant.

If a continuous map $f:S^d\to S^d$ has no fixed point, the formula $H_t(x)=((1-t)f(x)-tx)/\Vert (1-t)f(x)-tx\Vert$ is a homotopy from $f$ to the antipodal map. The denominator could vanish only at $t=1/2$ and $f(x)=x$, which is excluded. The homotopy invariance of mapping degree therefore gives $\mathrm{deg}f=(-1{)}^{d+1}$.

A free homotopy between maps is an ordinary homotopy with no specified basepoint held fixed. Two loops at the same basepoint are freely homotopic precisely when their elements of the fundamental group are conjugate: a homotopy traces a connecting basepoint path, and the boundary of its parameter square gives conjugation. Conversely that connecting path can be slid around the loop to realize the conjugation.

Suppose a finite group $G$ acts freely on $S^n$, $n>1$. The quotient has fundamental group $G$ and higher homotopy groups equal to those of $S^n$. Attach one $(n+1)$-cell killing a generator of ${\pi }_n\cong \mathbb{Z}$, then attach higher cells successively to kill all remaining higher homotopy. This constructs a Eilenberg–MacLane space $K(G,1)$ with only one $(n+1)$-cell. The one-cell bound on next-degree cohomology gives dimension at most one in degree $n+1$. But if $G=C_p\times C_p$, the mod-p cohomology ring of a cyclic group and Künneth theorem give dimension $n+2$ there. This contradiction excludes that group, and also excludes such a subgroup of any group acting freely on a sphere.

The fundamental group ${\pi }_1(X,x_0)$ consists of based loops modulo based homotopy. Multiplication follows one based loop and then the other; the constant based loop represents the identity, and path reversal gives the inverse. Passing to homotopy classes makes this operation associative, as explained by well-defined loop concatenation.

Identify the circle with $S^1=\{z\in \mathbb{C}:|z|=1\}$ and choose basepoint $1$. Its based loopshave winding number $n$: positive $n$ counts counterclockwise turns, negative $n$ counts clockwise turns, and $n=0$ gives the constant based loop. Every based loop on the circle is joined by a based homotopy to exactly one ${\gamma }_n$. ConsequentlyThis is Hatcher's Theorem 1.7 in Algebraic Topology, §1.1, pp. 29–30. For example, two counterclockwise turns followed by one clockwise turn represent $2+(-1)=1$, rather than three turns. The winding graph in the figure above makes the signed arithmetic visible.

A proof outline uses the covering map $q:\mathbb{R}\to S^1$, $q(u)=e^{2\pi iu}$. By the path lifting theorem, a based loop has a unique lift starting at zero and ending at an integer $n$. Interpolating its lift to $s\mapsto ns$ produces a based homotopy to ${\gamma }_n$. The homotopy lifting theorem for a covering map keeps that endpoint unchanged during a based homotopy, proving uniqueness.

In the filled disk, every based loop $\gamma$ at $x_0$ has a based homotopy to the constant based loop:Both endpoints remain $x_0$, and the convex set property of the disk keeps every intermediate point inside. Thus ${\pi }_1(D^2,x_0)$ is the trivial group. Together with the deformation retraction above, this computes three concrete answers: the filled disk has trivial fundamental group, while the circle and the annulus have infinite cyclic groups.

Maps have computable algebraic shadows too. The continuous map $p_n:S^1\to S^1$, $p_n(z)=z^n$, fixes $1$ and sends ${\gamma }_k$ to ${\gamma }_{nk}$. Its induced group homomorphism is therefore $k\mapsto nk$, and its mapping degree is $n$. For $n=2$ the image consists of even integers, so this map cannot be a homotopy equivalence. Composing $p_2$ with $p_{-3}$ gives $p_{-6}$: map composition multiplies degrees, whereas concatenating based loops adds winding numbers. Distinct degrees distinguish these maps up to homotopy; this calculation does not classify arbitrary topological spaces.

Historically, Poincaré's 1895 Analysis Situs introduced the fundamental group; its original §12 explicitly names it. The CNRS transcription of Poincaré's §12, Groupe fondamental provides a direct view of this early formulation.

The integral homology of the loop space of a sphere implies that the unit $S^{r-1}\to \Omega S^r$ has vanishing relative homology through degree $2r-3$. Both spaces are simply connected for $r\ge 3$. The absolute Hurewicz theorem first identifies their degree-two homotopy groups with homology; the relative Hurewicz theorem then rules out a first nonzero relative homotopy group in degrees three through $2r-3$. The relative homotopy sequence makes the unit an isomorphism on ${\pi }_i$ through degree $2r-4$. Loop-suspension adjunction identifies this map with suspension, yielding ${\pi }_i(S^{r-1})\cong {\pi }_{i+1}(S^r)$ in that range.

A homotopy between continuous maps $f,g:X\to Y$ is a continuous map $H:X\times [0,1]\to Y$ satisfying $H(x,0)=f(x)$ and $H(x,1)=g(x)$. Think of $t$ as time: continuity is required jointly in position and time. The target $Y$ specifies where every intermediate point is allowed to go. See Hatcher, Algebraic Topology, Chapter 0, pp. 3–4.

For the filled disk $D^2=\{x\in {\mathbb{R}}^2:\Vert x\Vert \le 1\}$, the formulais a contraction of a topological space: it starts at the identity map and ends at the constant map to zero. The Euclidean norm obeys $\Vert H(x,t)\Vert =(1-t)\Vert x\Vert \le 1$, so the formula never leaves the disk. For $x=(3/5,4/5)$, the midpoint of the deformation is $(3/10,2/5)$, whose Euclidean norm is $1/2$.

On the circle $S^1=\{x:\Vert x\Vert =1\}$, the same formula fails: at $t=1/2$ every image has Euclidean norm $1/2$, so lies outside $S^1$. This checks one failed candidate; the nonzero winding number of a once-around loop proves that no contraction within the circle exists. The distinction is between allowing the filling and requiring the deformation to stay on its boundary.